diff --git "a/8dFAT4oBgHgl3EQfpB03/content/tmp_files/load_file.txt" "b/8dFAT4oBgHgl3EQfpB03/content/tmp_files/load_file.txt" new file mode 100644--- /dev/null +++ "b/8dFAT4oBgHgl3EQfpB03/content/tmp_files/load_file.txt" @@ -0,0 +1,1386 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf,len=1385 +page_content='ERROR BOUNDS FOR KERNEL-BASED APPROXIMATIONS OF THE KOOPMAN OPERATOR FRIEDRICH PHILIPP, MANUEL SCHALLER, KARL WORTHMANN, SEBASTIAN PEITZ, AND FELIKS N ¨USKE ABSTRACT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' We consider the data-driven approximation of the Koopman operator for stochastic differen- tial equations on reproducing kernel Hilbert spaces (RKHS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Our focus is on the estimation error if the data are collected from long-term ergodic simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' We derive both an exact expression for the variance of the kernel cross-covariance operator, measured in the Hilbert-Schmidt norm, and probabilistic bounds for the finite-data estimation error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Moreover, we derive a bound on the prediction error of observables in the RKHS using a finite Mercer series expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Further, assuming Koopman-invariance of the RKHS, we provide bounds on the full approximation error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Numerical experiments using the Ornstein-Uhlenbeck process illustrate our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' INTRODUCTION The Koopman operator [23] has become an essential tool in the modeling process of complex dy- namical systems based on simulation or measurement data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' The philosophy of the Koopman approach is that for a (usually non-linear) dynamical system on a finite-dimensional space, the time-evolution of expectation values of observable functions satisfies a linear differential equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Hence, after “lifting” the dynamical system into an infinite-dimensional function space of observables, linear methods become available for its analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' The second step is then to notice that traditional Galerkin approximations of the Koopman operator can be consistently estimated from simulation or measurement data, establishing the fundamental connection between the Koopman approach and modern data science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Koopman methods have found widespread application in system identification [4], control [24, 42, 25, 17, 49], sensor place- ment [31], molecular dynamics [50, 44, 35, 36, 18, 56], and many other fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' We refer to [19, 33, 5] for comprehensive reviews of the state of the art.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' The fundamental numerical method for the Koopman approach is Extended Dynamic Mode Decom- position (EDMD) [54], which allows to learn a Galerkin approximation of the Koopman operator from finite (simulation or measurement) data on a subspace spanned by a finite set of observables, often called dictionary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' An appropriate choice of said dictionary is a challenging problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' In light of this issue, representations of the Koopman operator on large approximation spaces have been considered in recent years, including deep neural networks [29, 32], tensor product spaces [21, 37], and reproducing kernel Hilbert spaces (RKHS) [55, 11, 20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' In the work [20] it was shown that by means of the integral operator associated to an RKHS, it is possible to construct a type of Galerkin approximation of the Koopman operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' The central object are (cross-)covariance operators, which can be estimated from data, using only evaluations of the feature map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Due to the relative simplicity of the resulting numerical algorithms on the one hand, and the rich approximation properties of reproducing kernels on the other hand, kernel methods have emerged as a promising candidate to overcome the fundamental problem of dictionary selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' A key question is the quantification of the estimation error for (compressed1) Koopman operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' For finite dictionaries and independent, identically distributed (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=') samples, error estimates were provided in [26, 38], see also [58] for the ODE case and [49] for an extension to control-affine systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' The 1A compression of a linear operator T to a subspace M is given by PT|M, where P denotes a projection onto M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='08637v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='DS] 20 Jan 2023 2 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' PHILIPP, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' SCHALLER, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' WORTHMANN, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' PEITZ, AND F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' N ¨USKE estimation error for cross-covariance operators on kernel spaces was considered in [34], where general concentration inequalities were employed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' The data were also allowed to be correlated, and mixing coefficients were used to account for the lack of independence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' In this article, we take a different route and follow the approach of our previous paper [38], where we, in addition, also derived error estimates for the Koopman generator and operator for finite dictionaries and data collected from long-term, ergodic trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' This setting is relevant in many areas of science, where sampling i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' from an unknown stationary distribution is practically infeasible, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=', in fluid or molecular dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' The centerpiece of our results was an exact expression for the variance of the finite-data estimator, which can be bounded by an asymptotic variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' The asymptotic variance by itself is a highly interesting dynamical quantity, which can also be described in terms of Poisson equations for the generator [27, Section 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' We consider the Koopman semigroup (Kt)t≥0 generated by a stochastic differential equation on the space L2 µ, where µ is a probability measure which is invariant w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' the associated Markov process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' We study the action of Kt on observables in an RKHS H which is densely and compactly embedded in L2 µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' If this action is considered through the “lens” of the kernel integral operator E : L2 µ → H (see Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='2), we arrive at a family of operators Ct H = EKtE∗ (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' The action of Ct H : H → H is that of a cross-covariance operator: Ct Hψ = � (Ktψ)(x)k(x, ·) dµ(x), ψ ∈ H, where k(·, ·) is the kernel generating the RKHS H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' These operators possess canonical empirical estima- tors based on finite simulation data, which only require evaluations of the feature map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' L2 µ L2 µ H H Kt E E∗ Ct H FIGURE 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Diagram illustrating the different operators involved Our contribution, illustrated in Figure 2, is two-fold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' In our first main result, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='1, we provide an exact formula for the Hilbert-Schmidt variance of the canonical empirical estimator �Cm,t H of the cross- covariance operator Ct H, for m data points sampled from a long ergodic simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' This result extends the findings in [38] to the kernel setting and no longer depends on the dictionary size (which would be infinite, at any rate).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Due to the infinite-dimensional setting, additional assumptions are required, in particular, a spectral decomposition of the Koopman generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Our result allows for probabilistic estimates for the error ∥ �Cm,t H − Ct H∥HS, see Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' As a second main result, we propose an empirical estimator for the restriction of the Koopman op- erator Kt to H, truncated to finitely many terms of its estimated Mercer series expansion, and prove a probabilistic bound for the resulting estimation error in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='1, measured in the operator norm for bounded linear maps from H to L2 µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' This result can be seen as a bound on the prediction error for the RKHS-based Koopman operator due to the use of finite data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' In the situation where the RKHS is invariant under the Koopman operator we are able to complement the preceding error analysis with a bound on the full approximation error in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Finally, we illustrate our results for a one-dimensional Ornstein-Uhlenbeck (OU) process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' For this simple test case, all quantities appearing in our error estimates are known analytically and can be well approximated numerically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Therefore, we are able to provide a detailed comparison between the error bound obtained from our results and the actual errors observed for finite data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Our experiments show that ERROR BOUNDS FOR KERNEL-BASED APPROXIMATIONS OF THE KOOPMAN OPERATOR 3 our bounds for the estimation error of the cross-covariance operator are accurate, and that the corrections we introduced to account for the inter-dependence of the data are indeed required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Concerning the prediction error, we find our theoretical bounds still far too conservative, which reflects the problem of accounting for the effect of inverting the mass matrix in traditional EDMD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' This finding indicates that additional research is required on this end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Full Koopman Approximation Error Projection error Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='5 Variance representation of empirical estimator Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='1 Estimation error Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='1 i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' sampling Ergodic sampling Cross-covariance bound Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='4 ∥Ct ℍ − ̂ C m,t ℍ ∥HS ∥Kt N − ̂ K m,t N ∥ℍ→L2μ(X) ∥Kt − ̂ K m,t N ∥ℍ→L2μ(X) ∥Kt − Kt N∥ℍ→L2μ(X) FIGURE 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Illustration of main results The paper is structured as follows: the setting is introduced in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' The result concerning the variance of the empirical cross-covariance operator, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='1, is presented and proved in Section 3, while our bound for the prediction error is part of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='1 in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Numerical experiments are shown in Section 5, conclusions are drawn in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' PRELIMINARIES In this section, we provide the required background on stochastic differential equations (Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='1), reproducing kernel Hilbert spaces (Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='2), Koopman operators (Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='3), and their representa- tions on an RKHS (Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Stochastic differential equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Let X ⊂ Rd and let a stochastic differential equation (SDE) with drift vector field b : X → Rd and diffusion matrix field σ : X → Rd×d be given, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=', dXt = b(Xt) dt + σ(Xt) dWt, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='1) where Wt is d-dimensional Brownian motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' We assume that both b and σ are Lipschitz-continuous and that (1 + ∥ · ∥2)−1[∥b∥2 + ∥σ∥F ] is bounded on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Then [39, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='1] guarantees the existence of a unique solution (Xt)t≥0 to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' The solution (Xt)t≥0 constitutes a continuous-time Markov process whose transition kernel will be denoted by ρt : X ×BX → R, where BX denotes the Borel σ-algebra on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Then ρt(x, ·) is a probability measure for all x ∈ X, and for each A ∈ BX we have that ρt(·, A) is a representative of the conditional probability for A containing Xt given X0 = · , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=', ρt(x, A) = P(Xt ∈ A|X0 = x) for µ-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' x ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Throughout, we will assume the existence of an invariant (Borel) probability measure µ for the Markov process (Xt)t≥0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=', we have � ρt(x, A) dµ(x) = µ(A) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='2) for all t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' In addition to being invariant, we will often assume that µ is ergodic, meaning that for any t > 0 every ρt-invariant set A (that is, ρt(x, A) = 1 for all x ∈ A) satisfies µ(A) ∈ {0, 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' In this case, the Birkhoff ergodic theorem [15, Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='6] (see also (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='1)) and its generalizations apply, and allow us to calculate expectations w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' µ using long-time averages over simulation data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' We let ∥ · ∥p denote the Lp µ(X)-norm, 1 ≤ p < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' In the particular case p = 2, scalar product and norm on the Hilbert space L2 µ(X) will be denoted by ⟨· , ·⟩µ and ∥ · ∥µ, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' 4 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' PHILIPP, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' SCHALLER, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' WORTHMANN, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' PEITZ, AND F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' N ¨USKE 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Reproducing kernel Hilbert spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' In what follows, let k : X × X → R be a continuous and symmetric positive definite kernel, that is, we have k(x, y) = k(y, x) for all x, y ∈ X and m � i,j=1 k(xi, xj)cicj ≥ 0 for all choices of x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' , xm ∈ X and c1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' , cm ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' It is well known that k generates a so-called reproducing kernel Hilbert space (RKHS) [1, 6, 40] (H, ⟨· , ·⟩) of continuous functions, such that for ψ ∈ H the reproducing property ψ(x) = ⟨ψ, Φ(x)⟩, x ∈ X, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='3) holds, where Φ : X → H denotes the so-called feature map corresponding to the kernel k, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=', Φ(x) = k(x, ·), x ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' In the sequel, we shall denote the norm on H by ∥ · ∥ and the kernel diagonal by ϕ: ϕ(x) = k(x, x), x ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Then for x ∈ X we have ∥Φ(x)∥2 = ⟨Φ(x), Φ(x)⟩ = ⟨k(x, ·), k(x, ·)⟩ = k(x, x) = ϕ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' We shall frequently make use of the following estimate: |k(x, y)| = |⟨Φ(x), Φ(y)⟩| ≤ ∥Φ(x)∥∥Φ(y)∥ = � ϕ(x)ϕ(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' In particular, it shows that k is bounded if and only if its diagonal ϕ is bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' By Lp µ(X), p ∈ [1, ∞), we denote the space of all functions (not equivalence classes) on X with a finite p-norm ∥ · ∥p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Henceforth, we shall impose the following Compatibility Assumptions: (A1) ϕ ∈ L2 µ(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' (A2) If ψ ∈ L2 µ(X) such that � � k(x, y)ψ(x)ψ(y) dµ(x) dµ(y) = 0, then ψ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' (A3) If ψ ∈ H such that ψ(x) = 0 for µ-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' x ∈ X, then ψ(x) = 0 for all x ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Many of the statements in this subsection can also be found in [52, Chapter 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' However, as we aim to present the contents in a self-contained way, we provide the proofs in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' The following lemma explains the meaning of the compatibility assumptions (A1) and (A2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Under the assumption that ϕ ∈ L1 µ(X) (in particular, under assumption (A1)), we have that H ⊂ L2 µ(X) with ∥ψ∥µ ≤ � ∥ϕ∥1 · ∥ψ∥, ψ ∈ H, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='4) and assumption (A2) is equivalent to the density of H in L2 µ(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' We have meticulously distinguished between functions and equivalence classes as there might be distinct functions φ, ψ ∈ H, which are equal µ-almost everywhere2, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=', φ = ψ in L2 µ(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' The com- patibility assumption (A3) prohibits this situation so that H can in fact be seen as a subspace of L2 µ(X), which is then densely and continuously embedded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' 2For example, if µ = δa and φ(a) = ψ(a) ERROR BOUNDS FOR KERNEL-BASED APPROXIMATIONS OF THE KOOPMAN OPERATOR 5 Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' (a) Condition (A1) implies k ∈ L4 µ⊗µ(X × X), where µ ⊗ µ is the product measure on X × X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' (b) The density of H in L2 µ(X) is strongly related to the term universality in the literature, see [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' (c) Condition (A3) holds if supp µ = X, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' [52, Exercise 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' It immediately follows from � |ψ(x)|∥Φ(x)∥ dµ(x) ≤ ∥ψ∥µ∥ϕ∥1/2 1 , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='5) for ψ ∈ L2 µ(X) that the linear operator E : L2 µ(X) → H, defined by Eψ := � ψ(x)Φ(x) dµ(x), ψ ∈ L2 µ(X), is well defined (as a Bochner integral in H) and bounded with operator norm not larger than ∥ϕ∥1/2 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' The so-called kernel mean embedding Ek, mapping probability measures ν on X to the RKHS H, is defined by Ekν = � Φ(x) dν(x), see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=', [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Hence, we have Eψ = Ekν with dν = ψ dµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Note that the operator E is not an embedding in strict mathematical terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' The terminology embedding rather applies to its adjoint E∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Indeed, the operator E enjoys the simple but important property: ⟨Eψ, η⟩ = � ψ(x)⟨Φ(x), η⟩ dµ(x) = � ψ(x)η(x) dµ(x) = ⟨ψ, η⟩µ (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='6) for ψ ∈ L2 µ(X) and η ∈ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' This implies that the adjoint operator E∗ : H → L2 µ(X) is the inclusion operator from H into L2 µ(X), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=', E∗η = η, η ∈ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='7) We shall further define the covariance operator3 CH := EE∗ ∈ L(H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Recall that a linear operator T ∈ L(H) on a Hilbert space H is trace class if for some (and hence for each) orthonormal basis (ej)j∈N of H we have that �∞ j=1⟨(T ∗T)1/2ei, ei⟩ < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' A linear operator S ∈ L(H, K) between Hilbert spaces H and K is said to be Hilbert-Schmidt [12, Chapter III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='9] if S∗S is trace class, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=', ∥S∥2 HS := �∞ j=1 ∥Sei∥2 < ∞ for some (and hence for each) orthonormal basis (ej)j∈N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Let the Compatibility Assumptions (A1)–(A3) be satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Then the following hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' (a) The operator E is an injective Hilbert-Schmidt operator with ∥E∥2 HS = ∥ϕ∥1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' (b) The space H is densely and compactly embedded in L2 µ(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' (c) The operator CH is an injective non-negative selfadjoint trace class operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' The next theorem is due to Mercer and can be found in, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=', [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' It shows the existence of a particular orthonormal basis (ej)∞ j=1 of L2 µ(X) composed of eigenfunctions of E∗E, which we shall henceforth call the Mercer basis corresponding to the kernel k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Again for the sake of self-containedness, we give a short proof in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' 3In what follows, by L(H, K) we denote the set of all bounded (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=', continuous) linear operators between Hilbert spaces H and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' As usual, we also set L(H) := L(H, H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' 6 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' PHILIPP, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' SCHALLER, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' WORTHMANN, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' PEITZ, AND F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' N ¨USKE Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='5 (Mercer’s Theorem).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' There exists an orthonormal basis (ej)∞ j=1 of L2 µ(X) consisting of eigenfunctions of E∗E with corresponding eigenvalues λj > 0 such that �∞ j=1 λj = ∥ϕ∥1 < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Fur- thermore, (fj)∞ j=1 with fj = � λjej constitutes an orthonormal basis of H consisting of eigenfunctions of CH with corresponding eigenvalues λj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Moreover, for all x, y ∈ X, k(x, y) = � j fj(x)fj(y) = � j λjej(x)ej(y), the series converges absolutely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' The Koopman semigroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' The Koopman semigroup (Kt)t≥0 associated with the SDE (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='1) is defined by (Ktψ)(x) = E[ψ(Xt)|X0 = x] = � ψ(y) ρt(x, dy), for ψ ∈ B(X), the set of all bounded Borel-measurable functions on X, and ρt(x, dy) = dρt(x, ·)(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' It is easy to see that the invariance of µ is equivalent to the identity � Ktψ dµ = � ψ dµ (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='8) for all t ≥ 0 and ψ ∈ B(X) (which easily extends to functions ψ ∈ L1 µ(X), see Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Note that in the case σ = 0 the SDE (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='1) reduces to the deterministic ODE ˙x = b(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Then (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='8) implies � |ψ(φ(t, x))|2 dµ(x) = � |ψ(x)|2 dµ(x) for all t ≥ 0 and all ψ ∈ B(X), where φ(·, x) is the solution of the initial value problem ˙y = b(y), y(0) = x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Hence, the composition operator Kt : ψ �→ ψ ◦ φ(t, ·) is unitary in L2 µ(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' However, we shall require below (see Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='1) that Kt has its spectrum in the interior of the unit circle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Therefore, we assume throughout that σ ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' The proofs of the following two propositions can be found in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' For each p ∈ [1, ∞] and t ≥ 0, Kt extends uniquely to a bounded operator from Lp µ(X) to itself with operator norm ∥Kt∥Lp µ→Lp µ ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' By Cb(X) we denote the set of all bounded continuous functions on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' As the measure µ is finite, we have Cb(X) ⊂ B(X) ⊂ Lp µ(X) for all p ∈ [1, ∞].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' In fact, Cb(X) is dense in each Lp µ(X), p ∈ [1, ∞), see [48, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' (Kt)t≥0 is a C0-semigroup of contractions in Lp µ(X) for each p ∈ [1, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' The infinitesimal generator of the C0-semigroup (Kt)t≥0 is the (in general unbounded) operator in L2 µ(X), defined by Lψ = L2 µ- lim t→0 Ktψ − ψ t , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='9) whose domain dom L is the set of all ψ ∈ L2 µ(X) for which the above limit exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' By Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='8 and the Lumer-Phillips theorem (see [28]), the operator L is densely defined, closed4, dissipative (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=', Re⟨Lψ, ψ⟩µ ≤ 0 for all ψ ∈ dom L), and its spectrum is contained in the closed left half-plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' The constant function 1 is contained in dom L and L1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Moreover, if M := span{1} ⊂ L2 µ(X), then both M and M⊥ are invariant under L and all Kt, t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' 4Recall that a linear operator T, defined on a subspace dom T of a Hilbert space H, which maps to a Hilbert space K, is closed if its graph is closed in H × K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' ERROR BOUNDS FOR KERNEL-BASED APPROXIMATIONS OF THE KOOPMAN OPERATOR 7 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' It is easy to see that Kt1 = 1 for each t ≥ 0 and hence 1 ∈ dom L with L1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Hence KtM ⊂ M for all t ≥ 0 and LM ⊂ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Now, if ψ ⊥ 1, then ⟨Ktψ, 1⟩µ = � Ktψ dµ = � ψ dµ = ⟨ψ, 1⟩µ = 0, which shows that also KtM⊥ ⊂ M⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' The relation LM⊥ ⊂ M⊥ follows from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' □ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Representation of Koopman Operators on the RKHS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Using the integral operator E, it is possi- ble to represent the Koopman operator with the aid of a linear operator on H, which is based on kernel evaluations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' This construction mimics the well-known kernel trick used frequently in machine learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' To begin with, for any x, y ∈ X define the rank-one operator Cxy : H → H by Cxyψ := ��ψ, Φ(y)⟩Φ(x) = ψ(y)Φ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' For t ≥ 0 and ψ ∈ H we further define the cross-covariance operator Ct H : H → H by Ct Hψ := � � Cxyψ ρt(x, dy) dµ(x) = � (Ktψ)(x)Φ(x) dµ(x) = EKtψ = EKtE∗ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Thus, we have Ct H = EKtE∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='10) In other words, the cross-covariance operator Ct H represents the action of the Koopman semigroup through the lens of the RKHS integral operator E (see [20] for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Being the product of the two Hilbert-Schmidt operators EKt and E∗, the operator Ct H is trace class for all t ≥ 0 (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' [16, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' 521]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Note that due to ρ0(x, · ) = δx, for t = 0 this reduces to the already introduced covariance operator � � Cxy ρ0(x, dy) dµ(x) = � Cxx dµ(x) = EE∗ = CH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' The identity (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='10) shows that for all η, ψ ∈ H we have ⟨η, Ct Hψ⟩ = ⟨η, Ktψ⟩µ, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='11) which shows that the role of Ct H is analogous to that of the stiffness matrix in a traditional finite- dimensional approximation of the Koopman operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' In this analogy, the covariance operator CH plays the role of the mass matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Empirical estimators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Next, we introduce empirical estimators for Ct H based on finite data (xk, yk), k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' , m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' We consider two sampling scenarios for fixed t > 0: (1) The xk are drawn i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' from µ, and each yk ∼ µ is obtained from the conditional distribution ρt(xk, ·), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=', yk|(xk = x) ∼ ρt(x, ·) for µ-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' x ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' For example, yk can be obtained by simulating the SDE (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='1) starting from xk until time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' (2) µ is ergodic and both xk and yk are obtained from a single (usually long-term) simulation of the dynamics Xt at discrete integration time step ∆t > 0, using a sliding-window estimator, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=', x0 = X0 ∼ µ, xk = Xk∆t, and yk = Xk∆t+t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Moreover, we assume that there exists a Riesz basis (ψj)∞ j=0 of L2 µ(X) consisting of eigenfunc- tions of the generator L with corresponding eigenvalues µj satisfying �∞ j=0 e2(Re µj)∆t < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' It easily follows from the discussion in Appendix B that the last assumption on the generator L and on the decay of its eigenvalues µj is equivalent to the similarity of L to an (unbounded) normal operator N such that eN∆t ∈ L(L2 µ(X)) is Hilbert-Schmidt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' If the assumption holds with ψj = Sej, where (ej) is an orthonormal basis of L2 µ(X), the operator N is given by N = � j µj⟨ · , ej⟩ej with dom N = {ψ : (µj⟨ψ, ej⟩) ∈ ℓ2} and L = SNS−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' The condition �∞ j=0 e2(Re µj)∆t < ∞ then obviously means that the eigenvalues of eN∆t form an ℓ2 sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' 8 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' PHILIPP, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' SCHALLER, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' WORTHMANN, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' PEITZ, AND F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' N ¨USKE Recall that the joint distribution of two random variables X and Y is given by dPX,Y (x, y) = dPY |X=x(y) · dPX(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Set X = xk and Y = yk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Then, in both cases (1) and (2), we have PX = µ and PY |X=x(B) = P(yk ∈ B|xk = x) = P(Xt ∈ B|X0 = x) = ρt(x, B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' In other words, for the joint distribution µ0,t of xk and yk we have dµ0,t(x, y) = dρt(x, ·)(y) · dµ(x) = ρt(x, dy) · dµ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' More explicitly, µ0,t(A × B) = � A ρt(x, B) dµ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Now, since Ct H = � � Cxy ρt(x, dy) dµ(x) = � Cxy dµ0,t(x, y) = E � Cxk,yk � , for the empirical estimator for Ct H we choose the expression �Cm,t H = 1 m m−1 � k=0 Cxk,yk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='12) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' VARIANCE OF THE EMPIRICAL ESTIMATOR In case (1), the law of large numbers [3, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='4] and, in case (2), ergodicity [2] ensures the expected behavior lim m→∞ ∥ �Cm,t H − Ct H∥HS = 0 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' However, this is a purely qualitative result, and nothing is known a priori on the rate of this convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' The main result of this section, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='1, contains an exact expression for the Hilbert-Schmidt vari- ance of the empirical estimator �Cm,t H based on m data points, which then yields probabilistic estimates for the expression ∥ �Cm,t H − Ct H∥HS, see Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Here, our focus is on the estimation from a single ergodic trajectory, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=', case (2) above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' While the broader line of reasoning partially resembles that of our previous paper [38], we require additional steps due to the infinite-dimensional setting introduced by the RKHS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' In Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='1 and its proof, we will be concerned with evolving kernels kt : X × X → R, defined by kt(x, x′) := � � k(y, y′) ρt(x, dy) ρt(x′, dy′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' We have kt(x, x′) = � � ⟨Φ(y), Φ(y′)⟩ ρt(x, dy) ρt(x′, dy′) = �� Φ(y) ρt(x, dy), � Φ(y′) ρt(x′, dy′) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' The integrals in the last expression are well defined as limits in H for µ-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' x, x′ ∈ X as � � ∥Φ(y)∥ ρt(x, dy) dµ(x) = � � � ϕ(y) ρt(x, dy) dµ(x) = � � ϕ(x) dµ(x) ≤ ∥ϕ∥1/2 1 , see (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' This shows that kt is well defined ((µ ⊗ µ)-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=') and that it is a positive definite kernel on its domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Moreover, k0 = k and |kt(x, x′)| ≤ � � ϕ(y′) � � ϕ(y) ρt(x, dy) ρt(x′, dy′) = (Kt√ϕ)(x) · (Kt√ϕ)(x′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' ERROR BOUNDS FOR KERNEL-BASED APPROXIMATIONS OF THE KOOPMAN OPERATOR 9 In particular, kt ∈ L2 µ⊗µ(X 2) with ∥kt∥L2 µ⊗µ ≤ ∥ϕ∥1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' By Φt we denote the corresponding feature map, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=', Φt(x) = kt(x, ·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Note that not necessarily Φt(x) ∈ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Finally, we define Φt,x := Φ(x)Φt(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' We are now in the position to formulate our first main result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Setting zk = (xk, yk), k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' , m, the Hilbert-Schmidt variance of the empirical estimator can be written as E � ∥ �Cm,t H − Ct H∥2 HS � = 1 m � E0(t) + 2 m−1 � k=1 m−k m E � ⟨Czk − Ct H, Cz0 − Ct H⟩HS � � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='1) where E0(t) := E � ∥Cz0 − Ct H∥2 HS � = ⟨Ktϕ, ϕ⟩µ − ⟨k, kt⟩L2 µ⊗µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' In case (1), E � ∥ �Cm,t H − Ct H∥2 HS � = 1 mE0(t), whereas in case (2) we have E � ∥ �Cm,t H − Ct H∥2 HS � = 1 m � �E0(t) + 2 ∞ � j=1 dj,tqj 1 − qj � 1 − 1 m · 1 − qm j 1 − qj �� � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='2) with qj = eµj∆t, dj,t = ⟨cj,t, ψj⟩µ, and cj,t(x) = ⟨Φt,x, �ψj⟩µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Before proving Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='1 in Subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='1 below, let us comment on its statements and draw some conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' (a) Note that, by ergodicity of the invariant measure µ, the generator L has no eigenvalues on the imaginary axis, except the simple zero eigenvalue (see Proposition D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='1 in the Appendix).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' In contrast, if we drop the ergodicity assumption, we have E � ∥ �Cm,t H − Ct H∥2 HS � = 1 m � �E0(t) + 2 ∞ � j=ν0 dj,tqj 1 − qj � 1 − 1 m · 1 − qm j 1 − qj �� � + m − 1 m ν0−1 � j=1 dj,t, where ν0 = #{j : µj ∈ 2πi ∆t Z} is the number of eigenvalues of L of the form 2kπi ∆t , k ∈ Z, counting multiplicities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Obviously, the last term does not decay to zero as m → ∞ if �ν0−1 j=1 dj,t ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' (b) The definition of cj,t requires Φt,x to be in L2 µ(X) for µ-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' x ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' This will in fact be proved in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='6 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' In the following, we let σ2 m := E0(t) + 2 ∞ � j=1 dj,tqj 1 − qj � 1 − 1 m · 1 − qm j 1 − qj � and σ2 ∞ := E0(t) + 2 ∞ � j=1 dj,tqj 1 − qj .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Then E � ∥ �Cm,t H − Ct H∥2 HS � = σ2 m m and σ2 m → σ2 ∞ as m → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Both infinite series converge absolutely as (qj) ∈ ℓ2 by assumption, and (dj,t) ∈ ℓ2 as shown in the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' We can therefore interpret σ2 ∞ as asymptotic variance of the estimator ˆCm,t H , similar to our previous results in [38, Lemma 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' An upper bound on the variance can be obtained as follows: 10 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' PHILIPP, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' SCHALLER, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' WORTHMANN, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' PEITZ, AND F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' N ¨USKE Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' In case (2), for all m ∈ N we have σ2 m ≤ ⟨Ktϕ, ϕ⟩µ � 1 + 4B Aδq ∥q∥ℓ2 � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='3) where A and B denote the lower and upper Riesz bounds of (ψj), respectively, q = (qj)∞ j=1 , and δq = inf j≥1 |1 − qj| > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' First of all, by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='6, E0(t) = ⟨Ktϕ, ϕ⟩µ − ⟨k, kt⟩L2 µ⊗µ ≤ ⟨Ktϕ, ϕ⟩µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' We have |1 − qj| ≥ δq and |qj| ≤ 1 for all j ≥ 1 and hence 1 |1 − qj| · ����1 − 1 m · 1 − qm j 1 − qj ���� ≤ 1 δq � 1 + 1 m m−1 � k=0 |qj|k � ≤ 2 δq .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' This and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='7) imply (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' □ Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' We have the following probabilistic bound on the estimation error: P � ∥Ct H − �Cm,t H ∥HS > ε � ≤ � � � � � � � � � � � � � σ2 m mε2 , in case (2), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='4) E0(t) mε2 , in case (1), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='5) 2 e − mε2 8∥k∥2∞ , in case (1) with bounded kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='6) In particular, the above also holds upon replacing the left-hand side by P � ∥EKtψ − �Cm,t H ψ∥ > ε � for ψ ∈ H, ∥ψ∥ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' The inequalities (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='4) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='5) are an immediate consequence of Markov’s inequality, applied to the random variable ∥Ct H − �Cm,t H ∥2 HS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' The inequality (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='6) follows from Ct H − �Cm,t H = 1 m �m−1 k=0 (Ct H − Czk), Hoeffding’s inequality for Hilbert space-valued random variables [43, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='5] (see also [30, Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='2]), and (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='6 below) ∥Ct H − Cxy∥HS ≤ ∥Ct H∥HS + ∥Cxy∥HS = � ⟨k, kt⟩L2 µ⊗µ + � ϕ(x)ϕ(y) ≤ 2∥k∥∞, since also ∥kt∥∞ ≤ ∥k∥∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' The estimate ∥EKtψ − �Cm,t H ψ∥ = ∥EKtE∗ψ − �Cm,t H ψ∥ = ∥(Ct H − �Cm,t H )ψ∥ ≤ ∥Ct H − �Cm,t H ∥HS finally yields the last claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' □ Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Under additional assumptions (boundedness of the kernel, mixing, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' ), other concen- tration inequalities than Markov’s, such as, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=', [3, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='12] (α-mixing) or [46, Th´eor`eme 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='1] (β-mixing), might lead to better estimates than (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Let t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Then Φt,x ∈ L2 µ(X) for µ-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' x ∈ X with ∥Φt,x∥2 µ ≤ ϕ(x)(Ktϕ)(x) · ⟨Ktϕ, ϕ⟩µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Moreover, for every t ≥ 0 we have ∥Cxy∥2 HS = ϕ(x)ϕ(y) and ∥Ct H∥2 HS = ⟨k, kt⟩L2 µ⊗µ = � � Φt,x(y) dµ(y) dµ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' ERROR BOUNDS FOR KERNEL-BASED APPROXIMATIONS OF THE KOOPMAN OPERATOR 11 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' We estimate |Φt,x(x′)|2 = |k(x, x′)kt(x, x′)|2 ≤ ϕ(x)ϕ(x′)(Kt√ϕ)2(x) · (Kt√ϕ)2(x′) ≤ ϕ(x)(Ktϕ)(x) · ϕ(x′)(Ktϕ)(x′), where we have applied Jensen’s inequality to (Kt√ϕ)(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' This proves the first inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Next, if (fj) ⊂ H denotes the Mercer basis corresponding to k, then ⟨Cxy, Cx′y′⟩HS = � i ⟨Cxyfi, Cx′y′fi⟩ = � i fi(y)fi(y′)k(x, x′) = k(x, x′)k(y, y′) This proves ∥Cxy∥2 HS = ϕ(x)ϕ(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Moreover, it yields ∥Ct H∥2 HS = ���� � Cxy dµ0,t(x, y) ���� 2 HS = � � k(x, x′)k(y, y′) dµ0,t(x, y) dµ0,t(x′, y′) = � � k(x, x′) �� � k(y, y′) ρt(x, dy) ρt(x′, dy′) � dµ(x′) dµ(x) = ⟨k, kt⟩L2 µ⊗µ, as claimed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' □ Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' First of all, we have E � ∥ �Cm,t H − Ct H∥2 HS � = E ���� 1 m m−1 � k=0 (Czk − Ct H) ��� 2 HS � = E � 1 m2 m−1 � k,ℓ=0 � Czk − Ct H, Czℓ − Ct H � HS � = E � 1 m2 m−1 � k=0 ∥Czk − Ct H∥2 HS + 2 m2 m−1 � k=0 m−1 � ℓ=k+1 � Czk − Ct H, Czℓ − Ct H � HS � = 1 mE � ∥Cz0 − Ct H∥2 HS � + 2 m2 m−1 � k=1 (m − k)E � ⟨Czk − Ct H, Cz0 − Ct H⟩HS � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' where we exploited that E[⟨Czk − Ct H, Czℓ − Ct H⟩HS] only depends on the difference ℓ − k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Let us compute the first term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Since E[Cz0] = Ct H and thus E[⟨Cz0, Ct H⟩HS] = ∥Ct H∥2 HS, E � ∥Cz0 − Ct H∥2 HS � = E � ∥Cz0∥2 HS � − ∥Ct H∥2 HS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' For ψ ∈ H we have ∥Cz0ψ∥2 = ∥ψ(y0)Φ(x0)∥2 = ψ(y0)2ϕ(x0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Using the Mercer basis (fi) ⊂ H corresponding to k in H (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='5), we obtain E � ∥Cz0∥2 HS � = E � � i ∥Cz0fi∥2� = E � � i fi(y0)2ϕ(x0) � = E[ϕ(x0)ϕ(y0)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Note that the latter equals (ϕ(x) = k(x, x) by definition) E[ϕ(x0)ϕ(y0)] = � ϕ(x) � ϕ(y) ρt(x, dy) dµ(x) = � ϕ(x)(Ktϕ)(x) dµ(x) = ⟨Ktϕ, ϕ⟩µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' We obtain E � ∥Cz0 − Ct H∥2 HS � = E[ϕ(x0)ϕ(y0)] − ⟨k, kt⟩L2 µ⊗µ = ⟨Ktϕ, ϕ⟩µ − ⟨k, kt⟩L2 µ⊗µ = E0(t) and thus (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Case (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' In this case, zk and zℓ are independent for k ̸= ℓ, so that E � ⟨Czk − Ct H, Czℓ − Ct H⟩HS � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' 12 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' PHILIPP, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' SCHALLER, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' WORTHMANN, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' PEITZ, AND F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' N ¨USKE Hence, the statement of the theorem for case (1) follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Case (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Here, the cross terms do not vanish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' In fact, E � ⟨Czk − Ct H, Cz0 − Ct H⟩HS � = E[⟨Czk, Cz0⟩HS] − ∥Ct H∥2 HS = E � � i ⟨Czkfi, Cz0fi⟩ � − ∥Ct H∥2 HS = E �� � i fi(yk)fi(y0) � k(xk, x0) � − ∥Ct H∥2 HS = E � k(yk, y0)k(xk, x0) � − ∥Ct H∥2 HS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Now, E � k(yk, y0)k(xk, x0) � = � � � � k(y′, y)k(x′, x) ρt(x′, dy′) ρk∆t(x, dx′) ρt(x, dy) dµ(x) = � � k(x, x′) �� � k(y, y′) ρt(x, dy) ρt(x′, dy′) � ρk∆t(x, dx′) dµ(x) = � � k(x, x′)kt(x, x′) ρk∆t(x, dx′) dµ(x) = � �� [Φ(x)Φt(x)](x′) ρk∆t(x, dx′) � dµ(x) = � [Kk∆tΦt,x](x) dµ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Hence, E � ⟨Czk − Ct H, Cz0 − Ct H⟩HS � = � (Kk∆tΦt,x)(x) dµ(x) − ⟨k, kt⟩L2 µ⊗µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Let us now exploit the assumptions on the spectral properties of the generator L in case (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' For µ-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' x ∈ X, we have Φt,x = ∞ � j=0 cj,t(x)ψj, the series converging in L2 µ(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Therefore, KsΦt,x = ∞ � j=0 cj,t(x)Ksψj = ∞ � j=0 cj,t(x)eµjsψj, and thus (for k ≥ 1) � � Kk∆tΦt,x � (x) dµ(x) = � ∞ � j=0 cj,t(x)eµjk∆tψj(x) dµ(x) = ∞ � j=0 dj,t · eµjk∆t = ∞ � j=0 dj,t · qk j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' This series converges absolutely for each t ≥ 0 due to our assumption that � j |qj|2 < ∞ and since for each j ∈ N0 we have by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='6 that ∞ � j=0 |dj,t|2 ≤ B2 ∞ � j=0 ∥cj,t∥2 µ = B2 � ∞ � j=0 |⟨Φt,x, �ψj⟩µ|2 dµ(x) ≤ B2 A2 � ∥Φt,x∥2 µ dµ(x) ≤ B2 A2 ⟨Ktϕ, ϕ⟩2 µ, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='7) where A and B are the Riesz bounds of (ψj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' ERROR BOUNDS FOR KERNEL-BASED APPROXIMATIONS OF THE KOOPMAN OPERATOR 13 Without loss of generality, we may assume that µ0 = 0 with ψ0 = 1 and µ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' , µν0−1 ∈ 2πi ∆t Z, and ψk ∈ 1⊥ for k ≥ 1, see Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' The duality relations then imply �ψ0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Now, c0,t(x) = ⟨Φt,x, 1⟩µ = � k(x, y)kt(x, y) dµ(y) and hence d0,t = ⟨c0,t, 1⟩µ = � c0,t(x) dµ(x) = � � k(x, y)kt(x, y) dµ(y) dµ(x) = ⟨k, kt⟩L2 µ⊗µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='8) This implies E � ⟨Czk − Ct H, Cz0 − Ct H⟩HS � = ∞ � j=0 dj,t · qk j − ⟨k, kt⟩L2 µ⊗µ = ∞ � j=1 dj,t · qk j and therefore E � ∥ �Cm,t H − Ct H∥2 HS � = 1 mE0(t) + 2 m ∞ � j=1 dj,t m−1 � k=1 (1 − k m)qk j = 1 m � �E0(t) + 2 ν0−1 � j=1 dj,t m−1 � k=1 (1 − k m)qk j + 2 ∞ � j=ν0 dj,t m−1 � k=1 (1 − k m)qk j � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' The identity m−1 � k=1 � 1 − k m � qk = � q 1−q � 1 − 1 m · 1−qm 1−q � if q ̸= 1 m−1 2 if q = 1 finally yields (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' BOUND ON THE KOOPMAN PREDICTION ERROR The kernel cross-covariance operator Ct H can also be used to approximate the predictive capabilities of the Koopman operator, for observables in H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Approximating the full Koopman operator involves the inverse of the co-variance operator, which becomes an unbounded operator on a dense domain of definition in the infinite-dimensional RKHS case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Moreover, its empirical estimator �Cm H is finite-rank and thus not even injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' While Fukumizu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' tackle this problem in [10] by means of a regularization procedure, we choose to use pseudo-inverses instead (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' We truncate the action of the Koopman operator using N terms of the Mercer series expansion and derive a bound for the prediction error for fixed truncation parameter N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' While we use similar ideas as presented in [11], we heavily rely on our new results on the cross-covariance operator, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Afterwards, we deal with the case of Koopman-invariance of the RKHS [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Here, we establish an estimate for the truncation error, which then yields a bound on the deviation from the full Koopman operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' We emphasize that this error bound is extremely useful in comparison to its prior counterparts based on the assumption that the space spanned by a finite number of so-called observables (dictionary) is invariant under the Koopman operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' The latter essentially requires to employ only Koopman eigenfunctions as observables, see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=', [25, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Let (ej) be the Mercer orthonormal basis of L2 µ(X) corresponding to the kernel k and let λj = ∥Eej∥µ as well as fj := � λjej (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' We arrange the Mercer eigenvalues in a non-increasing way, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=', λ1 ≥ λ2 ≥ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Let ψ ∈ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Then Ktψ = ∞ � j=1 ⟨Ktψ, ej⟩µej = ∞ � j=1 ⟨Ct Hψ, ej⟩ej = N � j=1 ⟨Ct Hψ, ej⟩ej + ∞ � j=N+1 ⟨Ct Hψ, ej⟩ej.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='1) 14 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' PHILIPP, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' SCHALLER, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' WORTHMANN, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' PEITZ, AND F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' N ¨USKE 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Estimation error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' In the next theorem, we estimate the probabilistic error between the first sum- mand Kt Nψ = N � j=1 ⟨Ct Hψ, ej⟩ej, ψ ∈ H, and its empirical estimator, which is of the form �N j=1⟨ �Cm,t H ψ, �ej⟩�ej with approximations �ej of the ej.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Assume that the eigenvalues λj of CH are simple, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=', λj+1 > λj for all j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Fix an arbitrary N ∈ N and let δN = min j=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=',N λj − λj+1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='2) Further, let ε ∈ (0, δN) and δ ∈ (0, 1) be arbitrary and fix some5 m ≥ max{N, 2σ2 m ε2δ }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Let now �λ1 ≥ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' ≥ �λm denote the largest m eigenvalues of �Cm H in descending order and let �e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' , �em be corresponding eigenfunctions, respectively, such that ∥�ej∥ = �λ−1/2 j for j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' , m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' If we define �Km,t N ψ = N � j=1 ⟨ �Cm,t H ψ, �ej⟩�ej, ψ ∈ H, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='3) then, with probability at least 1 − δ, we have that ∥Kt N − �Km,t N ∥H→L2µ(X) ≤ � 1 √λN + N + 1 δNλN (1 + ∥ϕ∥1)∥ϕ∥1/2 1 � ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='4) All of the above statements equally apply to case (1) upon replacing σm by E0(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' (a) If we set �fj = �λ1/2 j �ej, then �Cm H = m � j=1 �λj⟨ · , �fj⟩ �fj, and thus N � j=1 ⟨ · , �ej⟩�ej = N � j=1 1 �λj ⟨ · , �fj⟩ �fj = ( �Cm H )† �QN, where �QN = �N j=1⟨ · , �fj⟩ �fj is the orthogonal projector onto the span of the first N eigenfunctions of �Cm H in H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Therefore, �Km,t N ψ = m � j=1 ⟨ �Cm,t H ψ, �ej⟩�ej = ( �Cm H )† �QN �Cm,t H ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' In particular, for N = m we have �Km,t N = ( �Cm H )† �Cm,t H , which surely is one of the first canonical choices for an empirical estimator of Kt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' (b) The functions �ej have unit length in the empirical L2 µ-norm: 1 m m � k=1 �ej(xk)�ej(xk) = � �Cm H �ej, �ej � = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Therefore, projecting onto the first N empirical Mercer features is the whitening transformation com- monly used in traditional EDMD [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' 5By Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='3, an amount of at least m ≥ max � N , 2∥ϕ∥2 µ ε2δ � 1 + 4B Aδq ∥q∥ℓ2 �� data points suffices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' ERROR BOUNDS FOR KERNEL-BASED APPROXIMATIONS OF THE KOOPMAN OPERATOR 15 Proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' By Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='4, both events ∥Ct H − �Cm,t H ∥HS ≤ ε and ∥CH − �Cm H ∥HS ≤ ε occur with probability at least 1 − δ/2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Hence, they occur simultaneously with probability at least 1 − δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' In the remainder of this proof we assume that both events occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Then all the statements deduced in the following hold with probability at least 1 − δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Let us define the intermediate approximation �Km,t N ψ = N � j=1 ⟨ �Cm,t H ψ, ej⟩ej, ψ ∈ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Let ψ ∈ H be arbitrary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Setting C := Ct H − �Cm,t H , we have ∥Kt Nψ − �Km,t N ψ∥2 µ = ����� N � j=1 � Cψ, ej � ej ����� 2 µ = N � j=1 ��� Cψ, ej ���2 = N � j=1 ��� ψ, C∗ej ���2 ≤ ∥ψ∥2 N � j=1 ∥C∗ej∥2 ≤ ∥ψ∥2 N � j=1 1 λj ∥C∗fj∥2 ≤ ∥ψ∥2 λN N � j=1 ∥C∗fj∥2 ≤ ∥ψ∥2 λN ∞ � j=1 ∥C∗fj∥2 = ∥ψ∥2 λN ∥Ct H − �Cm,t H ∥2 HS, and thus, ∥Kt Nψ − �Km,t N ψ∥µ ≤ ∥ψ∥ √λN ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Next, we aim at estimating the remaining error �Km,t N ψ − �Km,t N ψ = N � j=1 ⟨ �Cm,t H ψ, ej⟩ej − N � j=1 ⟨ �Cm,t H ψ, �ej⟩�ej = N � j=1 λ−1 j ⟨ �Cm,t H ψ, fj⟩fj − N � j=1 �λ−1 j ⟨ �Cm,t H ψ, �fj⟩ �fj = N � j=1 λ−1 j ⟨f, fj⟩fj − N � j=1 �λ−1 j ⟨f, �fj⟩ �fj = N � j=1 � λ−1 j Pjf − �λ−1 j �Pjf � = N � j=1 λ−1 j (Pj − �Pj)f + N � j=1 (λ−1 j − �λ−1 j ) �Pjf, where f = �Cm,t H ψ, Pjf = ⟨f, fj⟩fj and �Pjf = ⟨f, �fj⟩ �fj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' By (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='4), it suffices to estimate the above error in the ∥ · ∥-norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' By Theorem C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='3, the first summand can be estimated as ��� N � j=1 λ−1 j (Pj − �Pj)f ��� ≤ N � j=1 1 λj ∥Pj − �Pj∥∥f∥ ≤ N · ∥CH − �Cm H ∥ λNδN ∥f∥ ≤ N λNδN ∥f∥ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' 16 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' PHILIPP, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' SCHALLER, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' WORTHMANN, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' PEITZ, AND F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' N ¨USKE For the second summand we have ��� N � j=1 (λ−1 j − �λ−1 j ) �Pjf ��� 2 = N � j=1 |λ−1 j − �λ−1 j |2∥ �Pjf∥2 = N � j=1 |λj − �λj|2 λ2 j�λ2 j ∥ �Pjf∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Now, note that ϵ < δN by assumption and therefore ∥CH − �Cm H ∥HS ≤ δN ≤ λN−λN+1 2 ≤ λN 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' For j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' , N, according to Theorem C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='1 this implies �λj ≥ λj − |λj − �λj| ≥ λj − ∥CH − �Cm H ∥HS ≥ λj − λN 2 ≥ λj 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Hence, ��� N � j=1 (λ−1 j − �λ−1 j ) �Pjf ��� 2 ≤ 4 N � j=1 |λj − �λj|2 λ4 j ∥ �Pjf∥2 ≤ 4∥CH − �Cm H ∥2 HS λ4 N ∥ �QNf∥2, and thus, ��� N � j=1 (λ−1 j − �λ−1 j ) �Pjf ��� ≤ 2 λ2 N ∥f∥ε ≤ 1 λNδN ∥f∥ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' From ∥ �Cm,t H ∥ ≤ ∥ �Cm,t H − Ct H∥ + ∥Ct H∥ ≤ ∥ �Cm,t H − Ct H∥HS + ∥EKtE∗∥ ≤ ε + ∥ϕ∥1 we conclude �� �Km,t N ψ − �Km,t N ψ �� ≤ N + 1 λNδN ∥ �Cm,t H ψ∥ε ≤ N + 1 λNδN (ε + ∥ϕ∥1)∥ψ∥ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' All together, we obtain (recall (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='4)) ∥Kt Nψ − �Km,t N ψ∥µ ≤ ∥Kt Nψ − �Km,t N ψ∥µ + ∥ϕ∥1/2 1 ∥ �Km,t N ψ − �Km,t N ψ∥ ≤ ∥ψ∥ √λN ε + N + 1 λNδN (ε + ∥ϕ∥1)∥ϕ∥1/2 1 ∥ψ∥ε = � 1 √λN + N + 1 δNλN (1 + ∥ϕ∥1)∥ϕ∥1/2 1 � ε · ∥ψ∥, which implies (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Projection error in case of Koopman-invariance of the RKHS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' In the preceeding section, we have seen that the empirical operator �Km,t N can be written as ( �Cm H )† �Cm,t H if m = N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' In the limit m → ∞, we would arrive at the operator C−1 H Ct H, which is not even well-defined for all ψ ∈ H, in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' However, if the RKHS is invariant under Kt, the above operator limit is well-defined as a bounded operator on H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' In this situation we are able to extend Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='1 to an estimate on the full error made by our empirical estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' We start by defining the operator Kt H := C−1 H Ct H on its natural domain dom Kt H := {ψ ∈ H : Ct Hψ ∈ ran CH}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='5) We consider Kt H as an operator from H into itself (with domain of definition in H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' We have dom Kt H = {ψ ∈ H : Ktψ ∈ H}, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='6) and Kt H is closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' ERROR BOUNDS FOR KERNEL-BASED APPROXIMATIONS OF THE KOOPMAN OPERATOR 17 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Note that Ct Hψ ∈ ran CH if and only if EKtψ = CHφ for some φ ∈ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Since CHφ = Eφ and ker E = {0}, the latter is equivalent to Ktψ = φ ∈ H, which proves the representation of the domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' As to the closedness of Kt H, let (ψn) ⊂ dom Kt H and φ ∈ H such that ψn → ψ in H and Kt Hψn → φ in H as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' The latter implies Ct Hψn → CHφ, while the first implies Ct Hψn → Ct Hψ in H as n → ∞, from which we conclude that Ct Hψ = CHφ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=', ψ ∈ dom Kt H and Kt Hψ = φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' □ If the Koopman operator leaves the RKHS H invariant (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=', KtH ⊂ H), Kt H is defined on all of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Moreover, since the canonical inclusion map E∗ : H → L2(µ) is injective, it possesses an unbounded inverse on its range H, and therefore: C−1 H Ct Hφ = C−1 H EKtE∗φ = (EE∗)−1EE∗(E∗)−1KtE∗φ = (E∗)−1KtE∗φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='7) Remarkably, invariance of H under the Koopman operator implies that the left-hand side not only repro- duces the Koopman operator on H, but actually defines a bounded operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Parts of the next proposition can be found in [22, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='3] and [8, Theorem 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' For t > 0, the following statements are equivalent: (i) KtH ⊂ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' (ii) Kt H ∈ L(H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' (iii) ran Ct H ⊂ ran CH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' With regard to the two representations (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='5) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='6) of the domain, it is immediate that both (i) and (iii) are equivalent to dom Kt H = H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' The equivalence of the latter to (ii) follows from the closed graph theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' □ Note that if one of (i)–(iii) holds, then Kt H = Kt|H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' In addition to the assumptions in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='1, assume that H is invariant under the Koopman operator Kt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' For fixed N ∈ N, let δN be as in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='2), choose ε, δ, and m as in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='1 and define the empirical estimator �Km,t N as in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Then, with probability at least 1 − δ we have that ∥Kt − �Km,t N ∥H→L2µ(X) ≤ � λN+1 ∥Kt H∥ + � 1 √λN + N + 1 δNλN (1 + ∥ϕ∥1)∥ϕ∥1/2 1 � ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='8) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' First of all, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='1 implies that ∥Kt − �Km,t N ∥H→L2µ(X) ≤ ∥Kt − Kt N∥H→L2µ(X) + ∥Kt N − �Km,t N ∥H→L2µ(X) ≤ ∥Kt − Kt N∥H→L2µ(X) + � 1 √λN + N + 1 δNλN (1 + ∥ϕ∥1)∥ϕ∥1/2 1 � ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Now, for ψ ∈ H, ∥Ktψ − Kt Nψ∥2 µ = ����� ∞ � j=N+1 ⟨Ct Hψ, ej⟩ej ����� 2 µ = ∞ � j=N+1 |⟨Ct Hψ, ej⟩|2 = ∞ � j=N+1 1 λj |⟨Ct Hψ, fj⟩|2 = ∞ � j=N+1 1 λj |⟨Kt Hψ, CHfj⟩|2 = ∞ � j=N+1 λj|⟨Kt Hψ, fj⟩|2 ≤ λN+1∥Kt Hψ∥2, which proves the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' □ We have just proved that the projection error ∥Ktψ − Kt Nψ∥µ decays at least as fast as the square roots of the eigenvalues of CH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Recall that (λj)j∈N ∈ ℓ1(N), since CH is trace class with �∞ j=1 λj = Tr(CH) = ∥E∗∥2 HS = ∥ϕ∥1, see Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='4(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' 18 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' PHILIPP, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' SCHALLER, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' WORTHMANN, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' PEITZ, AND F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' N ¨USKE 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' ILLUSTRATION WITH THE ORNSTEIN-UHLENBECK PROCESS For the numerical illustration of our results, we consider the Ornstein-Uhlenbeck (OU) process on X = R, which is given by the SDE dXt = −αXt dt + dWt, where α > 0 is a positive parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Analytical Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Since all relevant properties of the OU process are available in analytical form, we can exactly calculate all of the terms appearing in our theoretical error bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Moreover, we can also compute the exact estimation and prediction errors for finite data in closed form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Let us begin by recapping the analytical results required for our analysis, which can be found in [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' The invariant measure µ, and the density of the stochastic transition kernel ρt, are given by dµ(x) = �α π e−αx2 dx and dρt(x, y) = � α πv2 t exp � − α v2 t (y − e−αtx)2� dx dy, with v2 t = (1 − e−2αt)/2α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' The Koopman operators Kt are self-adjoint in L2 µ(R), their eigenvalues and corresponding eigenfunctions are given by qj = e−αjt and ψj(x) = 1 � 2jαjj!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Hj( √ 2αx), j ∈ N0, where Hj are the physicist’s Hermite polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' We consider the Gaussian radial basis function (RBF) kernel with bandwidth σ > 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=', k(x, y) = exp � −(x − y)2 σ2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Let us quickly verify that this choice of the kernel satisfies the compatibility assumptions (A1)–(A3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Indeed, (A1) is trivial as k(x, x) = 1 and (A3) follows easily from the continuity of the functions in H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' To see that H is dense in L2 µ(R) (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=', (A2)), let ψ ∈ L2 µ(R) be such that ⟨ψ, Φ(y)⟩µ = 0 for all y ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' The latter means that φ∗ϕσ = 0, where φ(x) = ψ(x)e−αx2 and ϕσ(x) = e−x2/σ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' We apply the Fourier transform and obtain �φ · � ϕσ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Noting that the Fourier transform of a Gaussian is again a Gaussian, we get �φ = 0 and thus ψ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' The Mercer eigenvalues and features with respect to the invariant measure µ of the OU process, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=', the eigenvalues and eigenfunctions of the integral operator E∗E in L2 µ(R), are also available in analytical form [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' They are given by λi = � α C1 � 1 σ2C1 �i and ϕi(x) = γie−ζ2x2Hi �√αηx � , i ∈ N0, using the following constants: η = � 1 + 4 ασ2 �1/4 , γi = � η 2iΓ(i + 1) �1/2 , ζ2 = α 2 (η2 − 1), C1 = α + ζ2 + σ−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' With these results, we can compute the variance of the empirical estimator for Ct H as described in The- orem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' The eigenvalues qj were already given above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' The coefficients dj,t can be calculated using Mercer’s theorem as dj,t = � � k(x, x′)k(y, y′)ψj(x)ψj(x′) dµ0,t(x, y) dµ0,t(x′, y′) ERROR BOUNDS FOR KERNEL-BASED APPROXIMATIONS OF THE KOOPMAN OPERATOR 19 = � k,ℓ λkλℓ �� ϕk(x)ϕℓ(y)ψj(x) dµ0,t(x, y) �2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' The series needs to be truncated at a finite number of terms and the integrals can be calculated by numer- ical integration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' As d0,t = ⟨k, kt⟩L2 µ⊗µ = ∥Ct H∥2 HS (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='8)), and hence ∥Ct H∥2 HS = � k,ℓ λkλℓ �� ϕk(x)ϕℓ(y) dµ0,t(x, y) �2 , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='1) the Hilbert-Schmidt norm of the cross-covariance operator Ct H can be computed similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Since, for the Gaussian RBF kernel, we have ϕ(x) = k(x, x) = 1 for all x, we therefore find E0(t) = � Ktϕ, ϕ � µ − ∥Ct H∥2 HS = 1 − ∥Ct H∥2 HS, completing the list of terms required by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' In addition, we notice that upon replacing ei- ther one or two of the integrals in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='1) by finite-data averages, we can also calculate ∥ ˆCm,t H ∥2 HS and ⟨Ct H, ˆCm,t H ⟩HS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Therefore, the estimation error for finite data {(xk, yk)}m k=1 can be obtained by simply expanding the inner product ∥Ct H − ˆCm,t H ∥2 HS = ∥Ct H∥2 HS + ∥ ˆCm,t H ∥2 HS − 2⟨ ˆCm,t H , Ct H⟩HS, allowing us to precisely compare the estimation error to the error bounds obtained in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Besides the estimation error for Ct H, we are also interested in the prediction error, which is bounded according to Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' We will compare these bounds to the actual error ∥(Kt N − ˆKm,t N )φ∥L2µ(X), for a specific observable φ ∈ H and a fixed number of N Mercer features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' For the OU process, it is again beneficial to consider Gaussian observables φ: φ(x) = 1 � 2πσ2 0 exp � −(x − m0)2 2σ2 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Application of the Koopman operator leads to yet another, unnormalized Gaussian observable, which is given by Ktφ(x) = 1 � 2πσ2 t exp � −(m0 − e−αtx)2 2σ2 t � , σ2 t = σ2 0 + v2 t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' The inner products of Ktφ with the Mercer eigenfunctions ϕi can be evaluated by numerical integration, providing full access to the truncated observable Kt Nφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' On the other hand, the empirical approximation ˆKm,t N φ can be computed directly based on the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' We note that ˆKm,t N φ = N � j=1 � ˆCm,t H φ, ˆej � ˆej = 1 m m � k=1 φ(yk) N � j=1 ⟨Φ(xk), ˆej⟩ ˆej = 1 m m � k=1 φ(yk) N � j=1 ˆej(xk)ˆej.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' The functions ˆej can be obtained from the eigenvalue decomposition of the standard kernel Gramian matrix 1 mKX := 1 m [k(xk, xl)]m k,l=1 , as the latter is the matrix representation of the empirical covariance operator ˆCm H on the subspace span{Φ(xk)}m k=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' If 1 mKX = V ΛV ⊤ is the spectral decomposition of the Gramian, then ˆej = 1 m1/2ˆλj m � l=1 VljΦ(xl) 20 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' PHILIPP, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' SCHALLER, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' WORTHMANN, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' PEITZ, AND F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' N ¨USKE are the correctly normalized eigenfunctions according to Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Plugging this into the above, we find ˆKm,t N φ(x) = 1 m m � k=1 φ(yk) N � j=1 1 m1/2ˆλj m � l=1 Vljk(xl, xk) 1 m1/2ˆλj m � r=1 Vrjk(xr, x) = 1 mφ(Y )⊤ 1 mKX � VNΛ−2 N V ⊤ N � KX,x = 1 mφ(Y )⊤VNΛ−1 N V ⊤ N KX,x, where φ(Y ) = [φ(yk)]m k=1, KX,x = [k(xk, x)]m k=1, VN = V [IN 0m−N]⊤, ΛN = diag(�λj)N j=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Numerical Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' For the actual numerical experiments, we set α = 1, choose the elementary integration time step as ∆t = 10−2, and set the lag time to t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' We compute the exact variance E[∥Ct H− ˆCm,t H ∥2 HS] by the expression given in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='1, and also the coarser estimate for the variance given in Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' We test three different kernel bandwidths, σ ∈ {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='05, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='5}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' All Mercer series are truncated after the first 10 terms for σ ∈ {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='5}, and 20 terms for σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='05, while Koopman eigenfunction expansions are truncated after 15 terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' In the first set of experiments, we use Chebyshev’s inequality to compute the maximal estimation error ∥Ct H − ˆCm,t H ∥HS that can be guaranteed with confidence 1 − δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='9, for a range of data sizes m between m = 20 and m = 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' As a comparison, we generate 200 independent simulations of length m + t ∆t , corresponding to the sliding-window estimator with m data points, for each data size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' We then compute the resulting estimation error using the expressions given in the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' We extract the 1 − δ-percentile of the estimation error for all trajectories, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=', the maximal error that is not exceeded by 100 ∗ (1 − δ) percent of the trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' In addition, we also use Chebyshev’s inequality with the i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' variance 1 mE0(t) to predict the estimation error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' The comparison of these results for all data sizes m and the different kernel bandwidths is shown in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' We observe that the bound from Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='1 is quite accurate, over-estimating the actual error by about a factor three, and captures the detailed qualitative dependence of the estimation error on m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' The coarser bound from Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='3, however, appears to discard too much information, it over-estimates the error by one to two orders of magnitude, and also does not capture the initial slope for small m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Finally, we note that for the larger kernel bandwidths, the i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' variance is indeed too small, leading to an under-estimation of the error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' This observation confirms that it is indeed necessary to take the effect of the correlation between data points into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' In a second set of experiments, we test the performance of our theoretical bounds concerning the prediction of expectations for individual observables, obtained in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' For the same three Gaussian RBF kernels as in the first set of experiments, we consider the observable φ = ϕ0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=', the first Mercer feature, and choose N = 10 in the Mercer series expansion Kt Nφ and its empirical approximation ˆKm,t N φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Note that φ is a different observable depending on the bandwidth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Again, we set 1 − δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='9, and use the bound from Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='1 to bound the L2 µ-error between Kt Nφ and ˆKm,t N φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' As a comparison, we compute the actual L2 µ-error by numerical integration, using the fact that we can evaluate Kt Nφ and ˆKm,t N φ based on the discussion above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' We repeat this procedure 15 times and provide average errors and standard deviations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' The results for all three kernels are shown in Figure 4, and we find that our theoretical bounds are much too pessimistic in all cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' This finding highlights our previous observation that bounding the prediction error outside the RKHS still requires more in-depth research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' ERROR BOUNDS FOR KERNEL-BASED APPROXIMATIONS OF THE KOOPMAN OPERATOR 21 102 103 104 m 10 2 10 1 100 101 (m) Error for Ct , t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='05, = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='050 T 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='1 C 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='3 i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Data 102 103 104 m 10 2 10 1 100 101 (m) Error for Ct , t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='05, = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='100 T 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='1 C 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='3 i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Data 102 103 104 m 10 2 10 1 100 101 (m) Error for Ct , t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='05, = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='500 T 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='1 C 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='3 i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Data FIGURE 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Probabilistic error estimates for Ct H associated to the OU process, at lag time t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='05, and the Gaussian RBF kernel with different bandwidths σ ∈ {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='05, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='05} (corresponding to left, center and right panels).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' The blue and green curves show the es- timated error using the fine and coarse bounds from Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='1 and Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='3, re- spectively, while the purple curves represent the bound obtained from the i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='-variance 1 mE0(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' The red curve shows the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='9-percentile of the estimation error based on 200 independent simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' 102 103 m 10 1 101 103 105 107 Prediction Error t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='05, = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='050 Prediction N = 10 Data Bound N = 10 102 103 m 10 1 101 103 105 107 Prediction Error t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='05, = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='100 Prediction N = 10 Data Bound N = 10 102 103 m 10 1 101 103 105 107 Prediction Error t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='05, = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='500 Prediction N = 10 Data Bound N = 10 FIGURE 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Comparison of the theoretical bound on the prediction error ∥Kt Nφ − ˆKm,t N φ∥µ, if φ is chosen as the first Mercer feature ϕ0, using N = 10 in the Mercer series representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' The predicted error is shown in blue, error bars for the actual error obtained from 15 independent data sets are shown in red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Different panels correspond to the same kernel bandwidths as in Figure 3 above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' CONCLUSIONS We have analyzed the finite-data estimation error for data-driven approximations of the Koopman operator on reproducing kernel Hilbert spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' More specifically, we have provided an exact expression for the variance of empirical estimators for the cross-covariance operator, if a sliding-window estimator is applied to a long ergodic trajectory of the dynamical system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' This setting is relevant for many complex systems, such as molecular dynamics simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Our results present a significant improvement over the state of the art, since they concern a setting where the notorious problem of dictionary selection can be circumvented, and therefore no longer depend on the dictionary size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' We have also extended the concept of asymptotic variance to an infinite-dimensional approximation space for the Koopman operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Our numerical study on the Ornstein Uhlenbeck process has shown that, even using a simple mass concentration inequality, accurate bounds on the estimation error can be obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' In our second result, we have extended our estimates to a uniform bound on the prediction error for observables in the RKHS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Thereby, we have circumvented dealing with an unbounded inverse of the covariance operator by applying a finite-dimensional truncation of the associated Mercer series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' In case of Koopman-invariance of the RKHS, however, we were able to find a bound on the truncation error which then yields estimates for the full approximation error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' 22 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' PHILIPP, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' SCHALLER, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' WORTHMANN, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' PEITZ, AND F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' N ¨USKE Still, the resulting error bounds have proven very conservative in the numerical examples.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' WORTHMANN, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' PEITZ, AND F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' N ¨USKE APPENDIX A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' PROOFS Proof of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Let ψ ∈ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Then (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='4) follows from � |ψ(x)|2 dµ(x) = � |⟨ψ, Φ(x)⟩|2 dµ(x) ≤ ∥ψ∥2 � ϕ(x) dµ(x) = ∥ψ∥2∥ϕ∥1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Assume that (A2) holds and that ψ ∈ L2 µ(X) is such that ⟨ψ, Φ(x)⟩µ = 0 for all x ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Then 0 = � ⟨ψ, Φ(x)⟩µψ(x) dµ(x) = � � k(x, y)ψ(x)ψ(y) dµ(x) dµ(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Hence, ψ = 0 by (A2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Conversely, assume that H is dense in L2 µ(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Let ψ ∈ L2 µ(X) such that � � k(x, y)ψ(x)ψ(y) dµ(x) dµ(y) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Since the integrand equals ⟨ψ(x)Φ(x), ψ(y)Φ(y)⟩ and the integral � ψ(x)Φ(x) dµ(x) exists by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='5), we obtain � ψ(x)Φ(x) dµ(x) = 0H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' This implies that ⟨ψ, Φ(y)⟩µ = � ψ(x)k(x, y) d��(x) = 0 for each y ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Hence, ⟨ψ, φ⟩µ = 0 for each φ ∈ H := span{Φ(x) : x ∈ X}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Now, let φ ∈ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Then there exists a sequence (φn) ⊂ H such that ∥φn − φ∥ → 0 as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Therefore, |⟨ψ, φ⟩µ| = |⟨ψ, φ − φn⟩µ| ≤ ∥ψ∥µ∥φ − φn∥µ ≤ ∥ψ∥µ � ∥ϕ∥1∥φ − φn∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Hence, ⟨ψ, φ⟩µ = 0, and the density of H in L2 µ(X) implies ψ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' □ Proof of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' (a) For ψ ∈ L2 µ(X) we have ∥Eψ∥2 = � � ψ(x)ψ(y)⟨Φ(x), Φ(y)⟩ dµ(x) dµ(y) = � � k(x, y)ψ(x)ψ(y) dµ(x) dµ(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Hence, the injectivity of E follows from (A2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' If (ei) is an orthonormal basis of H, then � i ∥E∗ei∥2 µ = � i ∥ei∥2 µ = � i � |ei(x)|2 dµ(x) = � i � |⟨Φ(x), ei⟩|2 dµ(x) = � ∥Φ(x)∥2 dµ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' The claim is now a consequence of ∥Φ(x)∥2 = ϕ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' (b) By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='1, H is dense in L2 µ(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Moreover, E∗ is compact by (a) and Schauder’s theorem [47, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' (c) This follows from (a) and ker CH = ker EE∗ = ker E∗ = {0} by (A3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' □ Proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='4, the operator E∗E ∈ B(L2 µ(X)) is a positive self-adjoint trace- class operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Hence, by the well known spectral theory of compact operators (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=', [12]) there exists an orthonormal basis (ej)∞ j=1 of L2 µ(X) consisting of eigenfunctions of E∗E corresponding to a summable sequence (λj)∞ j=1 of strictly positive eigenvalues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Since E∗ψ = ψ for ψ ∈ H, we have Eej = λjej and thus ej ∈ H for all j and CHej = EE∗ej = Eej = λjej.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Moreover, ⟨fi, fj⟩ = � λj/λi⟨Eei, ej⟩ = � λj/λi⟨ei, ej⟩µ = δij by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='6) so that the fj indeed form an orthonormal system in H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' The completeness of (fj) in H follows from the injectivity of E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Finally, �∞ j=1 λj = Tr CH = ∥ϕ∥1 and k(x, y) = ⟨Φ(x), Φ(y)⟩ = � j ⟨Φ(x), fj⟩⟨fj, Φ(y)⟩ = � j fj(x)fj(y), which completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' □ Proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Let ψ ∈ B(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' For p = ∞ we have |(Ktψ)(x)| = |Ex[ψ(Xt)]| ≤ Ex[|ψ(Xt)|] ≤ ∥ψ∥∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' If p < ∞, by Jensen’s inequality, for every convex φ : R → R we have φ ◦ Ktψ ≤ Kt(φ ◦ ψ) and thus |Ktψ|p ≤ Kt|ψ|p, which, by invariance of µ, leads to ∥Ktψ∥p p = � |Ktψ|p dµ ≤ � Kt|ψ|p dµ = � |ψ|p dµ = ∥ψ∥p p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' ERROR BOUNDS FOR KERNEL-BASED APPROXIMATIONS OF THE KOOPMAN OPERATOR 25 The claim now follows by density of B(X) in Lp µ(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' □ Proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Let ψ ∈ Cb(X) and fix x ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Denote the stochastic solution process of the SDE (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='1) with initial value x by Xx t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Since Xx t (ω) is continuous in t for P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' ω ∈ Ω (see [39, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='1]), ψ(Xx t (ω)) → ψ(Xx 0 (ω)) = ψ(x) as t → 0 for P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' ω ∈ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Hence, by dominated convergence, Ktψ(x) = E[ψ(Xx t )] = � ψ(Xx t (ω)) dP(ω) → ψ(x) as t → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' It now follows from Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='7 and, again, dominated convergence that ∥Ktψ −ψ∥p → 0 as t → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' If ψ ∈ Lp µ(X) and ε > 0, there exists η ∈ Cb(X) such that ∥ψ − η∥p < ε/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Choose δ > 0 such that ∥Ktη − η∥p < ε/3 for t < δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Then ∥Ktψ − ψ∥p ≤ ∥Kt(ψ − η)∥p + ���Ktη − η∥p + ∥η − ψ∥p < ε for t < δ, which proves the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' □ APPENDIX B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' RIESZ BASES Recall that a Riesz basis [7] of a Hilbert space H is a sequence (ψj) ⊂ H satisfying span{ψj} = H and for which there exist A, B > 0 such that for all c ∈ ℓ2, A∥c∥2 ≤ ��� � j cjψj ��� H ≤ B∥c∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' The constant A (B, resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=') is called a lower (upper, resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=') Riesz bound of the basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Also recall that to every Riesz basis (ψj) there exists a dual Riesz basis ( �ψj) such that ⟨ψj, �ψk⟩H = δjk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' If (ψj) has the bounds A and B, then ( �ψj) has bounds 1/B and 1/A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Every element f of H admits a representation f = � j⟨f, �ψj⟩Hψj = � j⟨f, ψj⟩H �ψj and A2∥f∥2 H ≤ � j |⟨f, ψj⟩|2 ≤ B2∥f∥2 H and B−2∥f∥2 H ≤ � j |⟨f, �ψj⟩|2 ≤ A−2∥f∥2 H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' It can furthermore be easily seen that a sequence (ψj) ⊂ H is a Riesz basis of H if and only if there exists a boundedly invertible linear operator S ∈ L(H) and an orthonormal basis (ej) of H such that ψj = Sej for all j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Then �ψj = (S−1)∗ej for all j, B = ∥S∥, and A = ∥S−1∥−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' APPENDIX C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' SOME FACTS FROM SPECTRAL THEORY In this section, let H be a Hilbert space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' If P is an orthogonal projection in H, we set P ⊥ = I − P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' For v ∈ H, ∥v∥ = 1, denote by Pv the rank-one orthogonal projection onto span{v}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' We say that a linear operator on H is non-negative if it is self-adjoint and its spectrum is contained in [0, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' For a non-negative compact operator T on H we denote by λ1(T) ≥ λ2(T) ≥ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' the eigenvalues of T in descending order (counting multiplicities).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' We set λj(T) = 0 if j > rank(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Moreover, if T has only simple eigenvalues6, we let Pj(T) denote the orthogonal projection onto the eigenspace ker(T − λj(T)) and Qn(T) = �n j=1 Pj(T) the spectral projection corresponding to the n largest eigenvalues of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Theorem C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='1 ([12, Cor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' If T and �T are two non-negative compact operators on H, then for all j ∈ N, |λj(T) − λj( �T)| ≤ ∥T − �T∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' 6i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=', dim ker(T − λ) = 1 for each eigenvalue λ of T 26 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' PHILIPP, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' SCHALLER, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' WORTHMANN, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' PEITZ, AND F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' N ¨USKE Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' For v, w ∈ H with ∥v∥ = ∥w∥ = 1 we have ∥Pv − Pw∥ = ∥P ⊥ w Pv∥ = � 1 − |⟨v, w⟩|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='1) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' First of all, the second equation in (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='1) is clear, since ∥P ⊥ w Pvf∥2 = ∥⟨f, v⟩P ⊥ w v∥2 = |⟨f, v⟩|2(1 − ∥Pwv∥2) = |⟨f, v⟩|2(1 − |⟨v, w⟩|2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Second, if Pv,w denotes the orthogonal projection onto Hv,w := span{v, w}, we have ∥Pv − Pw∥ = ∥(Pv − Pw)Pv,w∥ = ∥(Pv − Pw)|Hv,w∥ = sup x∈Hv,w, ∥x∥=1 ∥(Pv − Pw)x∥, which is a two-dimensional problem in Hv,w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Now, if x ∈ Hv,w, ∥x∥ = 1, we write x = av + bw and obtain a2 + 2abγ + b2 = 1, where γ = ⟨v, w⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Moreover, ⟨x, v⟩ = a + bγ, ⟨x, w⟩ = aγ + b and so ∥(Pv − Pw)x∥2 = ∥⟨x, v⟩v − ⟨x, w⟩w∥2 = ∥(a + bγ)v − (aγ + b)w∥2 = (a + bγ)2 − 2(a + bγ)(aγ + b)γ + (aγ + b)2 = a2 + 2abγ + b2γ2 − 2γ(a2γ + abγ2 + ab + b2γ) + a2γ2 + 2abγ + b2 = (1 − γ2)a2 + 2abγ − 2abγ3 + b2(1 − γ2) = (1 − γ2)(a2 + b2 + 2abγ) = 1 − |⟨v, w⟩|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Hence, the objective function is constant on {x ∈ Hv,w : ∥x∥ = 1} and (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='1) is proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' □ The next theorem is a variant of the Davis-Kahan sin(Θ) theorem (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' [57]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Theorem C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Let T and �T be non-negative Hilbert-Schmidt operators on H, let n ∈ N, assume that the largest n + 1 eigenvalues of T are simple, and set δ = min j=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=',n λj(T) − λj+1(T) 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' If ∥T − �T∥HS < δ, then for j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' , n we have ∥Pj(T) − Pj( �T)∥ ≤ ∥T − �T∥ δ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' For j ∈ N put λj = λj(T), Pj = Pj(T), �λj = λj( �T), and �Pj = Pj( �T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' By Theorem C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='1, we have |λj − �λj| ≤ ∥T − �T∥HS < δ for all j, hence �λj is contained in the interval Ij = (λj − δ, λj + δ) for j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' , n + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' By assumption, sup Ij+1 ≤ inf Ij for j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' In particular, the intervals I1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' , In+1 are pairwise disjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Now, let j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' , n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Then for k ∈ N \\ {j} we have |�λk − λj| > δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Therefore, we have dist(λj, σ( �T)\\{�λj}) ≥ δ and thus, for f ∈ �P ⊥ j H, ∥( �T − λj)f∥ ≥ dist � λj, σ( �T| �P ⊥ j H) � ∥f∥ = dist(λj, σ( �T)\\{�λj})∥f∥ ≥ δ∥f∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' As TPj = λjPj and �P ⊥ j �T = �T �P ⊥ j , we obtain ∥T − �T∥ ≥ ∥ �P ⊥ j ( �T − T)Pj∥ = ∥ �P ⊥ j �TPj − �P ⊥ j TPj∥ = ∥( �T − λj) �P ⊥ j Pj∥ ≥ δ∥ �P �� j Pj∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' The claim now follows from Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' □ ERROR BOUNDS FOR KERNEL-BASED APPROXIMATIONS OF THE KOOPMAN OPERATOR 27 APPENDIX D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' ERGODICITY AND THE GENERATOR In this section, we prove the following proposition on the spectral properties of the generator L under the ergodicity assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Proposition D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Assume that the invariant measure µ is ergodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Then ker L = span{1} and ker(L − iωI) = {0} for ω ∈ R\\{0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' First of all, it is worth mentioning that Lψ = 0 implies Ktψ = ψ for all t ≥ 0 and that Lψ = iωψ, ω ∈ R \\ {0}, implies K2π/ωψ = ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Therefore, it suffices to show that Ktψ = ψ for some t > 0 and ψ ∈ L2 µ(X) is only possible for constant ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' For this, we consider the Markov process (Xnt)∞ n=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' For convenience, we assume w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' that t = 1 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' By invariance of µ, the process (Xn)∞ n=0 is stationary, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=', (Xn)∞ n=0 and (Xn+1)∞ n=0 are equally distributed as X N0-valued random variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' According to [15, Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='2] there exist X-valued random variables X−k, k ∈ N, such that X := (Xn)n∈Z is also stationary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' By Pµ denote the law of the X Z-valued random variable X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' On S := X Z define the left shift T : S → S by T(xn)n∈Z := (xn+1)n∈Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Stationarity of X means that also TX ∼ Pµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' A set A ∈ BZ X := � k∈Z BX is called shift-invariant if T −1A = A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' It is easy to see that the set of shift-invariant sets forms a sub-σ-algebra I of BZ X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Now, by [13, Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='11] and the ergodicity of µ we have Pµ(A) ∈ {0, 1} for any A ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Now, Birkhoff’s Ergodic Theorem [15, Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='6] states that lim n→∞ 1 n n−1 � k=0 f(T kX) = E � f(X)|X−1I � (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='1) almost surely and in L1(Ω) for any f ∈ L1(S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Given ψ ∈ L1 µ(X), let us apply this theorem to the function f = ψ ◦ π0, where the projection π0 : S → X is defined by π0(xn)n∈Z = x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' First of all, � |f| dPµ = � |ψ(x0)| dPµ((xn)n∈Z) = � |ψ(x)| dµ(x) < ∞ as Pµ ◦ π−1 0 = µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Hence, we have f ∈ L1(S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Furthermore, we compute f(T kX) = ψ(π0(T kX)) = ψ(Xk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' For A ∈ I we have P(X−1A) = Pµ(A) ∈ {0, 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Thus, we obtain lim n→∞ 1 n n−1 � k=0 ψ(Xk) = E[f(X)] = � f dPµ = � ψ ◦ π0 dPµ = � ψ dµ almost surely and in L1(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Therefore, if ψ ∈ L2 µ(X) such that Ktψ = ψ, then Kktψ = ψ for all k ∈ N0, hence for µ-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' x ∈ X we have ψ(x) = 1 n n−1 � k=0 ψ(x) = 1 n n−1 � k=0 Kktψ(x) = 1 n n−1 � k=0 E[ψ(Xkt)|X0 = x] = E � 1 n n−1 � k=0 ψ(Xkt) ����� X0 = x � n→∞ −→ � ψ dµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Thus, ψ must indeed be (µ-essentially) constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' □ 28 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' PHILIPP, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' SCHALLER, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' WORTHMANN, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' PEITZ, AND F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' N ¨USKE AUTHOR AFFILIATIONS F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Philipp TECHNISCHE UNIVERSIT ¨AT ILMENAU, INSTITUTE FOR MATHEMATICS, WEIMARER STRASSE 25, D-98693 ILMENAU, GERMANY Email address: friedrich.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='philipp@tu-ilmenau.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='de M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Schaller TECHNISCHE UNIVERSIT ¨AT ILMENAU, INSTITUTE FOR MATHEMATICS, WEIMARER STRASSE 25, D- 98693 ILMENAU, GERMANY Email address: manuel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='schaller@tu-ilmenau.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='de K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Worthmann TECHNISCHE UNIVERSIT ¨AT ILMENAU, INSTITUTE FOR MATHEMATICS, WEIMARER STRASSE 25, D-98693 ILMENAU, GERMANY Email address: karl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='worthmann@tu-ilmenau.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='de S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' Peitz PADERBORN UNIVERSITY, DEPARTMENT OF COMPUTER SCIENCE, DATA SCIENCE FOR ENGINEERING, GER- MANY Email address: sebastian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='peitz@upb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='de F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content=' N¨uske MAX PLANCK INSTITUTE FOR DYNAMICS OF COMPLEX TECHNICAL SYSTEMS, MAGDEBURG, GERMANY Email address: nueske@mpi-magdeburg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='mpg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'} +page_content='de' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dFAT4oBgHgl3EQfpB03/content/2301.08637v1.pdf'}