diff --git "a/5tAyT4oBgHgl3EQfcfeN/content/tmp_files/load_file.txt" "b/5tAyT4oBgHgl3EQfcfeN/content/tmp_files/load_file.txt" new file mode 100644--- /dev/null +++ "b/5tAyT4oBgHgl3EQfcfeN/content/tmp_files/load_file.txt" @@ -0,0 +1,838 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf,len=837 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content='00284v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content='DG] 31 Dec 2022 SQUARE ROOT NORMAL FIELDS FOR LIPSCHITZ SURFACES AND THE WASSERSTEIN FISHER RAO METRIC EMMANUEL HARTMAN∗, MARTIN BAUER†, AND ERIC KLASSEN‡ Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' The Square Root Normal Field (SRNF) framework is a method in the area of shape analysis that defines a (pseudo) distance between unparametrized surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' For piecewise linear (PL) surfaces it was recently proved that the SRNF distance between unparametrized surfaces is equivalent to the Wasserstein Fisher Rao (WFR) metric on the space of finitely supported measures on S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' In the present article we extend this point of view to a much larger set of surfaces;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' we show that the SRNF distance on the space of Lipschitz surfaces is eqivalent to the WFR distance between Borel measures on S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' For the space of spherical surfaces this result directly allows us to characterize the non-injectivity and the (closure of the) image of the SRNF transform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' In the last part of the paper we further generalize this result by showing that the WFR metric for general measure spaces can be interpreted as an optimization problem over the diffeomorphism group of an independent background space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' Introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' The investigations of this article are motivated by applications in the area of mathematical shape analysis, which seeks to quantify differences, perform classification, and explain variability for populations of shapes [51, 40, 13, 28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' More specifically, the results of this article concern the Square Root Normal Field distance [16] on the space of surfaces and the Wasserstein Fisher Rao metric [9, 26] from unbalanced optimal transport.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' Before we describe the contributions of the current work in more detail, we will briefly summarize some results from these two areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' Shape analysis of surfaces: For the purpose of this article we consider a shape to be a parametrized surface or curve in Rd, where we identify two objects if they only differ by a trans- lation and/or a reparametrization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' In practice, it is often of interest to mod out by further shape preserving group actions, such as the groups of rotations or scalings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' To keep the presentation simple, we will ignore these additional finite dimensional groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' Consequently, the resulting shape space is an infinite dimensional, non-linear (quotient) space, which makes the application of statistical tech- niques to analyse these types of data a highly challenging task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' A common approach to overcome these difficulties can be found in the area of geometric statistics [35, 36], in which one develops statistical frameworks based on (Riemannian) geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' In the context of shape analysis of surfaces or curves, a variety of different metrics have been proposed for this purpose;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' this includes metrics induced by (right-invariant) metrics on diffeomorphism groups [51, 31] and reparametrization invariant metrics on the space of immersions [40, 3, 30], which are directly related to the investigations of the present article as we will explain next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' In the latter approach the calculation of the distance (similarity) between two shapes reduces to two tasks: calculating the geodesic distance on the space of immersions (parametrized surfaces or curves, resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=') and minimizing over the action of the shape preserving group actions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=', diffeomorphisms of the parameter space and translations in Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' In general there do not exist any explicit formulas for geodesics and thus computing solutions to the geodesic boundary value problems (and thus of the distance) is a highly non-trivial task and usually has to be solved using numerical optimization techniques, see eg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' [14, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' For specific examples of Riemannian metrics, however, simplifying transformations have been developed that allow for explicit calculations of geodesics and geodesic distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' This includes in particular the family of Ga,b-metrics on the space of curves [5, 34, 33, 50], a family of first order Sobolev type metrics, that are often called elastic metrics due to their connections to linear elasticity theory;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' see eg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' [33, 8, 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' For the specific choice of parameters a = 1, b = 1/2 the corresponding transformation is the so-called Square-Root-Velocity (SRV) transform [39], which is widely used in ∗Department of Mathematics, Florida State University (ehartman@fsu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content='edu) †Department of Mathematics, Florida State University and University of Vienna (bauer@math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content='fsu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content='edu) ‡Department of Mathematics, Florida State University (klassen@math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content='fsu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content='edu) 1 2 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' BAUER, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' HARTMAN, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' KLASSEN applications;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' see [40] and the references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' The advantage of this transformation is that it reduces the shape comparison problem to a single optimization over the shape preserving group actions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=', in the setting of the present article over reparametrizations and translations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' This computational simplification has led to both the development of efficient algorithms [49, 12, 39] and to analytic results on existence of minimizers and optimal parametrizations [7, 24, 44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' The family of elastic Ga,b metrics has a natural generalization to a four parameter family of metrics on the space of surfaces [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' Similarly to the case of curves, simplifying transformations have also been proposed in this more complicated situation [19, 20, 16, 41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' Notably, as a generalization of the SRV transform, the Square Root Normal Field (SRNF) transformation [16] has been introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' In contrast to the situation for curves, the corresponding Riemannian metric for this transformation is degenerate and, furthermore, it only leads to a first order approximation of the geodesic distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' Nonetheless it defines a reparametrization invariant (pseudo-) distance on the space of surfaces, which still allows for efficient computations using several methods of approximating the optimization over the diffeomorphism group [23, 4] and has proven successful in several applications, see [21, 17, 29, 22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' and the references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' Unbalanced Optimal transport: The second core theme of the present article can be found in the theory of optimal transport (OT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' Since Monge’s formulation of OT as a non-convex optimization problem in the space of transport maps, many formulations of the problem have been proposed to give insight to the theoretical properties of the problem as well as efficient methods for computing the solution, see [45, 46] for a comprehensive overview on the field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' In classical optimal transport theory one considers normalized (probability) distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' It is, however, important for many applications to relax this normalization assumption and compute trans- portation plans between arbitrary positive measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' Motivated by this observation the theory of optimal transport has been extended to measures with different masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' This field, called unbalanced optimal transport, has seen rapid developments in the past years and several different frameworks have been proposed [9, 25, 27, 37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' Among them is the Wasserstein Fisher Rao (WFR) distance, an interpolating distance between the quadratic Wasserstein metric and the Fisher–Rao metric, that was introduced independently by [9] and [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' The WFR distance has been applied to a variety of problems where it is more natural to consider optimal transport in an unbalanced setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' These applications range from color transfer [10], to earthquake epicenter location [52] and document semantic similarity metrics [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' Because of the growing field of applications, several algorithms have been proposed to compute the Wasserstein Fisher Rao metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' A variation on the popular Sinkhorn algorithm to solve for an entropy regularized version of the distance was proposed by [10] and an alternating minimization algorithm that computes an exact solution was introduced in [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' Contributions of the article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' Recently a new and surprising relationship between these two areas (shape analysis and unbalanced optimal transport) has been found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' Namely, in [6] it has been shown that for triangulated surfaces the calculation of the SRNF shape distance can be reduced to calculating the WFR distance between their corresponding surface area measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' The presentation in [6] was entirely focused on the discrete (PL) setting and the proof of the result essentially reduced to algebraic considerations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' In the first part of the present article we build the analytical tools to extend this result to the infinite dimensional setting, which contains in particular the original setup of the SRNF distance;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' the space of smooth surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' The main result of this part of our article – cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content='1 – shows that the SRNF shape distance between any two Lipschitz surfaces is equal to the WFR distance between their surface area measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' As a direct consequence of this result we are able to answer two fundamental questions regarding the SRNF transform: since the inception of the SRNF transform, it has been understood that the map is neither injective nor surjective [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' Characterizing the image and non-injectivity have, however, remained open problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' Recently a first degeneracy result in the context of closed surfaces has been found [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' Using our equivalence result we are able to obtain a characterization of the closure of the 3 image of this transform – cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content='6 – and a new strong degeneracy result of the corresponding distance (non-injectivity of the transform, resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=') – cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' In the second part we further explore the equivalence result for more general unbalanced optimal transport problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' Generalizations of some of the intermediate results of the first part allow us to offer a novel formulation of the WFR metric as a diffeomorphic optimization problem – cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' Whereas the main result of the first part of the article relates the WFR on S2 with a specific choice of parameter to a diffeomorphic optimization problem, we here extend this relationship to the WFR with any choice of parameter defined on any connected, compact, oriented Riemannian manifold, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' Notably, the space of diffeomorphisms we have to optimize over does not depend on N, but can be chosen as the diffeomorphism group of some background manifold, that only needs to be of dimension greater than or equal to two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' The authors thank FX Vialard and Cy Maor for useful discussions during the preparation of this manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' Bauer was supported by NSF-grants 1912037 and 1953244 and by FWF grant P 35813-N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' Hartman was supported by NSF grant DMS-1953244.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' Preliminaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' The Wasserstein Fisher Rao Distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' In the following, we will summarize the Kan- torovich formulation of the Wasserstein Fischer Rao distance, as introduced in [11] for measures on a smooth, connected, compact, oriented Riemannian manifold, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' Therefore we denote by M(N) the space of finite Borel measures on N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' In the Kantorovich formulation of the Wasserstein-Fisher-Rao distance, we will define a functional on the space of semi-couplings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' Therefore we first recall the definition of a semi-coupling: Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content='1 (Semi-couplings [11]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' Given µ, ν ∈ M(N) the set of all semi-couplings from µ to ν is given by Γ(µ, ν) = � (γ0, γ1) ∈ M(N × N)2|(Proj0)#γ0 = µ, (Proj1)#γ1 = ν � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' To define the Wasserstein-Fisher-Rao distance from µ to ν we define a functional on the space of semi-couplings from µ to ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' Let d denote the geodesic distance on N and δ ∈ (0, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' We consider the functional Jδ : Γ(µ, ν) → R (γ1, γ2) �→ 4δ2 � µ(N) + ν(N) − 2 � N×N √γ1γ2 γ (u, v)cos(d(u, v)/2δ)dγ(u, v) � where γ ∈ M(N × N) such that γ1, γ2 ≪ γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' Note that in the case where N = S2, we have d(u, v) = cos−1(u · v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' Thus for δ = 1 2, Jδ(γ1, γ2) = � S2×S2 ���� �γ1 γ (u, v)u − �γ1 γ (u, v)v ���� 2 dγ(u, v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content='1) Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content='2 (Wasserstein-Fisher-Rao Distance [11, 26]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' The Wasserstein-Fisher-Rao Dis- tance on M(N) is given by WFRδ : M(N) × M(N) → R≥0 defined via (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content='2) (µ, ν) �→ inf (γ0,γ1)∈Γ(µ,ν) � Jδ(µ, ν).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content='3) Some results in this article will specifically apply to the case where δ = 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' To simplify our notation, we define J := J1/2 and WFR := WFR1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' 4 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' BAUER, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' HARTMAN, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' KLASSEN 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' The Square Root Normal Field Shape Distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' In mathematical shape analysis, one defines metrics that measure the differences between geometric objects [51, 3, 40, 13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' In this article we consider geometric objects described by unparameterized surfaces which are elements of an infinite dimensional non-linear space modulo several finite and infinite dimensional group action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' As a result, computations in this space are difficult and even simple statistical operations are not well defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' Riemannian geometry can help to overcome these challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' In such a framework, one considers the space of all surfaces as an infinite dimensional manifold and equips it with a Riemannian metric that is invariant to the group action, which allows one to consider the induced metric on the quotient space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' For our purposes we will consider immersions of a smooth, connected, compact, oriented Rie- mannian 2-dimensional manifold, M, with or without boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' We denote the space of all Lipschitz immersions of M into R3 by Imm(M, R3), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=', Imm(M, R3) = {f ∈ W 1,∞(M, R3) : T f is inj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content='} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content='4) As we are interested in unparametrized surfaces, we have to factor out the action of the group of diffeomorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' In the context of Lipschitz immersions the natural group of reparametrizations for us to consider is the group of all orientation preserving, bi-Lipschitz diffeomorphisms: Γ(M) = {γ ∈ W 1,∞(M, M) : γ−1 ∈ W 1,∞(M, M), |Dγ| > 0 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' }, where |Dγ| denotes the Jacobian determinant of γ, which is well-defined as Dγ ∈ L∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' Note that this reparametrization group acts by composition from the right on Imm(M, R3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' In addition to the action by the reparametrization group, we also want to identify surfaces that only differ by a translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' This leads us to consider the following quotient space: S := Imm(M, R3)/(Γ(M) × trans) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content='5) In the following we will equip Imm(M) with a reparameterization invariant distance;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' the so called square root normal field (SRNF) distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' The SRNF map (distance resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=') was originally introduced by Jermyn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' in [15] for the space of smooth immersions, but it naturally extends to the space of all Lipschitz surfaces, as demonstrated in [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' We now recall the definition of this distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' For any given f ∈ Imm(M, R3), the orientation on M allows us to consider the unit normal vector field nf : M → R3, which is well-defined as an element of L∞(M, R3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' Furthermore, let {v, w} be an orthonormal basis of TxM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' Then for any f ∈ Imm(M, R3) we can define the area multiplication factor at x ∈ M via af(x) = |dfx(v) × dfx(w)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' The SRNF map is then given by Φ : Imm(M, R3)/ translations → L2(M, R3) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content='6) f �→ qf where qf(x) := � af(x) nf(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content='7) From this transform we define a distance on Imm(M, R3)/ translations by dImm(f1, f2) = ∥Φ(f1) − Φ(f2)∥L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' Next we consider a right-action of Γ(M) on L2(M, R3) that is compatible with the mapping Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' For q ∈ L2(M, R3) and γ ∈ Γ(M) we let (q ∗ γ)(x) = � |Dγ(x)|q(γ(x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content='8) It is easy to check that the action of Γ(M) on L2(M, R3) is by linear isometries and that for any f ∈ Imm and γ ∈ Γ, Φ(f) ∗ γ = Φ(f ◦ γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' 5 Thus, it follows that the SRNF distance on Imm(M, R3) is invariant with respect to this action and thus it descends to a (pseudo) distance on the quotient space S, which is given by dS([f1], [f2]) = inf γ∈Γ(M) d(f1, f2 ◦ γ), [f1], [f2] ∈ S(M) As we will see later the induced (pseudo) distance on the quotient space is highly degenerate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' Equivalence of WFR and SRNF in the piecewise linear category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' In [6] a surprising equivalence of the WFR and SRNF distance was shown: for piecewise linear surfaces it was proved that the SRNF distance can be reduced to the WFR distance between finitely supported measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' To formulate this result in detail we first associate to every q ∈ L2(M, R3) a measure on S2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' namely, for any open U ⊆ S2, we define q∗U = {x ∈ M|q(x) ̸= 0 and q(x)/|q(x)| ∈ U} and define the map L2(M, R3) → M(S2) via q �→ µq where for U ⊆ S2, µq(U) = � q∗U q(x) · q(x)dm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' The result proved in [6] is then formulated as: Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' Given two piecewise linear surfaces S1 and S2 parameterized by f and g, the SRNF shape distance can be computed as an unbalanced transport problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' More precisely, we have dS([f], [g]) = inf γ∈Γ(M) ∥qf − qg ∗ γ∥ = WFR(µqf , µqg).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' where qf and qg are the SRNFs of f and g respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' In the next section we will extend this result of to all Lipschitz immersions (Borel-measures, resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' The SRNF distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' For the goal of extending the result of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content='3 to all Lipschitz surfaces, we will consider specifically δ = 1 2 in the definition of the WFR metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' Equivalence of the WFR and SRNF distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' Our main result of this section is the following theorem, which is slightly stronger than the desired equivalence result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' Given q1, q2 ∈ L2(M, R3), inf γ∈Γ(M) ∥q1 − q2 ∗ γ∥L2 = WFR(µq1, µq2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' In particular, given f, g ∈ W 1,∞(M, R3) we can calculate their SRNF distance as an unbalanced OMT problem via dS([f], [g]) = WFR(µqf , µqg), where qf and qg are the SRNFs of f and g respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' Note, that as a direct consequence of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content='1 we can also conclude the extension of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content='3 to the original setup of the SRNF distance, the space of all smooth surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' The proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content='1 relies on a series of technical lemmas, which we will show next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' 6 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' BAUER, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' HARTMAN, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' KLASSEN Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' Let X, Y be topological spaces and ρ : X → Y be a measurable function with respect to the Borel σ-algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' If µ, µ1 ∈ M(X), γ, γ1 ∈ M(Y ) such that µ1 ≪ µ, γ = ρ∗µ, and γ1 = ρ∗µ1, then γ1 ≪ γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' Furthermore, µ1 µ = γ1 γ ◦ ρ almost everywhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' Let U ⊆ Y open such that γ(U) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' By definition, µ(ρ−1(U)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' Since µ1 ≪ µ, µ1(ρ−1(U)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' Therefore, γ1(U) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' This proves γ1 ≪ γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' Following the definitions of the Radon-Nikodym derivatives, pushforwards, and the change of variables formula, we obtain � ρ−1(U) µ1 µ dµ = � ρ−1(U) dµ1 = � U dγ1 = � U γ1 γ dγ = � ρ−1(U) γ1 γ ◦ ρ dµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' Thus, µ1 µ = γ1 γ ◦ ρ almost everywhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' Given q ∈ L2(M, R3) we can define a function from M to S2 that takes every point x ∈ M to the unit vector in the direction of q(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' As a matter of defining this function on every point, we can canonically choose the north pole of S2 for points where q(x) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' For q ∈ L2(M, R3) we define the unit vector map of q as q : M → S2 given by x �→ � q(x) |q(x)| if q(x) ̸= 0 (1, 0, 0) otherwise .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' Note that since q ∈ L2(M, R3), it follows that q : M → S2 is measurable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' Let q ∈ L2(M, R3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' We can define a measure, νq ∈ M(M), via νq(U) = � U |q(x)|2dm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' for all open U ⊆ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' Note that νq ≪ m and νq m = |q|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' Further, we can equivalently define µq as the pushforward of νq via q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' Let q ∈ L2(M, R3) and µq ∈ M(S2) be the measure associated with q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' Then µq = q∗νq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' Let U ⊆ S2 open and define M0 = {x ∈ M|q(x) = 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' If (1, 0, 0) ̸∈ S2, q−1(U) = q∗(U) and thus q∗νq(U) = � q−1(U) |q(x)|2dm = � q∗(U) |q(x)|2dm = µq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' If (1, 0, 0) ∈ S2, q−1(U) = q∗(U) ∪ M0 and thus q∗νq(U) = � q−1(U) |q(x)|2dm = � q∗(U) |q(x)|2dm + � M0 |q(x)|2dm = µq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' Leveraging what we have proven above we may show a key continuity result that will then allow us to complete the proof of the main theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' The map (L2(M, R3), ∥ · ∥L2) → (M(S2), WFR) defined via q �→ µq given by Equa- tion (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content='3) is Lipschitz continuous with Lipschitz constant K = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' 7 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' Let q1, q2 ∈ L2(M, R3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' For any semi-coupling (γ1, γ2) ∈ Γ(µq1, µq2), WFR(µq1, µq2) ≤ � Jδ(γ1, γ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' Thus, to prove the theorem we must construct (γ1, γ2) ∈ Γ(µq1, µq2) such that Jδ(γ1, γ2) = ∥q1−q2∥2 L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' To construct such a semi-coupling we first construct ρ : M → S2 × S2 defined as unit vector maps of q1 and q2 on the first and second factor respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' the map is given by ρ(x) = (q1(x), q2(x)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' Since q1 and q2 are individually measurable, then so is ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' We can then define γ1, γ2 ∈ M(S2 × S2) via γ1 = ρ∗νq1 and γ2 = ρ∗νq2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' Claim 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' The pair of measures, (γ1, γ2) is a semi-coupling from µq1 to µq2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' Proof of claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' Let U ⊆ S2 be open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' Thus, γ1(U × S2) = νq1 � ρ−1(U × S2) � = νq1 � q1−1(U) ∩ q2−1(S2) � = νq1 � q1−1(U) � = µq1(U) and γ2(S2 × U) = νq2 � ρ−1(S2 × U) � = νq1 � q1−1(S2) ∩ q2−1(U) � = νq1 � q2−1(U) � = µq2(U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' So (γ1, γ2) is a semi-coupling from µq1 to µq2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' Recall from the definition of the functional Jδ we need to construct γ ∈ M(S2 × S2) such that γ1, γ2 ≪ γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' Define γ = ρ∗m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' We know µq1, µq2 ≪ m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' Thus, by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content='2, γ1, γ2 ≪ γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' Furthermore, |q1|2 = µq1 m = γ1 γ ◦ ρ a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' and |q2|2 = µq2 m = γ2 γ ◦ ρ a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' So,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' Jδ(γ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' γ2) = � S2×S2 ���� �γ1 γ (u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' v)u − �γ1 γ (u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' v)v ���� 2 dγ(u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' v) = � S2×S2 γ1 γ (u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' v)dγ(u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' v) + � S2×S2 γ2 γ (u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' v)dγ(u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' v) − 2 � S2×S2 √γ1γ2 γ (u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' v)⟨u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' v⟩dγ(u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' v) = � ρ−1(S2×S2) γ1 γ ◦ ρ(x) dm + � ρ−1(S2×S2) γ2 γ ◦ ρ(x) dm − 2 � ρ−1(S2×S2) �γ1 γ ◦ ρ(x) �γ2 γ ◦ ρ(x)⟨ρ(x)⟩dγ(u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' v) = � M |q1(x)|2dm + � M |q2(x)|2dm − 2 � M |q1(x)||q2(x)| � q1(x) |q1(x)|,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' q2(x) |q2(x)| � dm =∥q1 − q2∥2 L2 Thus,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' WFR(µq1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' µq2) ≤ � Jδ(γ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' γ2) = 1 · ∥q1 − q2∥L2 We are now ready to conclude the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content='1: Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' Let q1, q2 ∈ L2(M, R3) and let ǫ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' Let p1, p2 be piecewise constant functions such that ∥q1 − p1∥L2 < ǫ/4 and ∥q2 − p2∥L2 < ǫ/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' Therefore, inf γ∈Γ(M) ∥q1 − p1 ∗ γ∥L2, inf γ∈Γ(M) ∥q2 − p2 ∗ γ∥L2, WFR(µq1, µp1), WFR(µq2, µp2) < ǫ/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' 8 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' BAUER, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' HARTMAN, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' KLASSEN Thus,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' inf γ∈Γ(M) ∥q1 − q2 ∗ γ∥L2 ≤ inf γ∈Γ(M) ∥q1 − p1 ∗ γ∥L2 + inf γ∈Γ(M) ∥p2 − q2 ∗ γ∥L2 + inf γ∈Γ(M) ∥p1 − p2 ∗ γ∥L2 ≤ ǫ/2 + inf γ∈Γ(M) ∥p1 − p2 ∗ γ∥L2 = ǫ/2 + WFR(µp1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' µp2) ≤ ǫ/2 + WFR(µq1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' µp1) + WFR(µp2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' µq2) + WFR(µq1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' µq2) ≤ ǫ + WFR(µq1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' µq2) and WFR(µq1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' µq2) ≤ WFR(µp1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' µp2) + WFR(µq1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' µp1) + WFR(µp2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' µq2) ≤ WFR(µp2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' µq2) + ǫ/2 = inf γ∈Γ(M) ∥p1 − p2 ∗ γ∥L2 + ǫ/2 ≤ inf γ∈Γ(M) ∥q1 − p1 ∗ γ∥L2 + inf γ∈Γ(M) ∥p2 − q2 ∗ γ∥L2 + inf γ∈Γ(M) ∥q1 − q2 ∗ γ∥L2 + ǫ/2 ≤ inf γ∈Γ(M) ∥q1 − q2 ∗ γ∥L2 + ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' So, WFR(µq1, µq2) − ǫ ≤ inf γ∈Γ(M) ∥q1 − q2 ∗ γ∥L2 ≤ WFR(µq1, µq2) + ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' Taking ǫ → 0 we can conclude infγ∈Γ(M) ∥q1 − q2 ∗ γ∥L2 = WFR(µq1, µq2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' Characterizing the closure of the image of the SRNF map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' Our equivalence result will also allow us to characterize the (closure of the) image of the SRNF map Φ in the context of spherical surfaces: Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' Let f ∈ Imm(S2, R3) and let q = Φ(f) ∈ L2(S2, R3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' Then q satisfies the closure condition � S2 q(x)|q(x)|dm = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' Moreover, the closure of the image of Φ is given by the set U := � q ∈ L2(S2, R3) such that � S2 q(x)|q(x)|dm = 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' To prove this result we will need a classical theorem from geometric measure theory and the study of convex polyhedra, which we will recall next: Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content='7 (Minkowski’s Theorem [1, 32, 38]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' Let µ ∈ M(S2) such that the support of µ is not concentrated on a great circle and � S2 x dµ(x) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' Then, there exists a unique (up to translation) convex body whose surface area measure is µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' Moreover, if µ is finitely supported then the convex body is a polytope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' 9 Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content='. Let f ∈ Imm(S2, R3) and qf = Φ(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' Let S = f(S2) and V be the surface enclosed by S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' Therefore, � S2 qf(x)|qf(x)|dm = � S2 af(x)nf(x)dm = � S nfdS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' Thus, this is the integral of the normal vector of a closed surface in R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' A simple application of the divergence theorem shows that the integral of the normal vector of the closed surface is zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' To see this, let {ei}3 i=1 be the unit basis vectors of R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' For i = 1, 2, 3, � S (nf · ei) dS = � V (∇ · ei) dV = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' Therefore, � S2 qf(x)|qf(x)|dm = 0 and the image of Φ is contained in U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' To prove the converse direction let q ∈ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' We aim to construct a convex body f with µqf arbitrarily close to µq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' By the definition of U the measure µq satisfies � S2 n dµq(n) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' Since finitely supported measures are dense with respect to the WFR metric, we can choose a finitely supported measure µq such that � S2 n dµq(n) = 0 and WFR(µq, µq) < ǫ/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' If the support of µq is not concentrated on a great circle we can invoke the Minkowski theorem and the result follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' For the general case we will slightly deform the measure as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' Define ˆµq := µq + 3 � i=1 ǫ 18δei + 3 � i=1 ǫ 18δ−ei where {ei}3 i=1 is the set of unit basis vectors of R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' Then ˆµq is a finitely supported measue and satisfies � S2 n d ˆµq(n) = 0 and ˆµq is not supported on a single great circle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' Moreover, WFR(µq, ˆµq) < ǫ/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' By the Minkowski Theorem (Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content='7) there exists a convex polytope with surface area measure given by ˆµq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' Let f ∈ W 1,∞(S2, R3) be the PL spherical parameterization of this convex body, so that µqf = ˆµq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' Thus, there exists γ ∈ Γ(M) such that ∥qf − q ∗ γ∥L2 < WFR(µqf , µq) + ǫ/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' Therefore, ∥qf − q ∗ γ∥L2 ≤ WFR(µqf , µq) + ǫ/3 = WFR( ˆµq, µq) + ǫ/3 ≤ WFR( ˆµq, µq) + WFR(µq, µq) + ǫ/3 < ǫ, which concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' Characterizing the degeneracy of the SRNF distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' As a second important con- sequence of the our equivalence result we can give a detailed proof of the degeneracy of the SRNF distance for smooth surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' Degeneracy results were studied in [18] and it was further characterized for certain PL surfaces in [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' Here we will generalize the characterization of [6] to smooth surfaces: Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' For any smooth, regular surface f ∈ C∞(S2, R3) ∩ Imm(S2, R3) there exists a unique (up to translations) convex body that is indistinguishable from f by the SRNF shape distance, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content='e, dS([f], [f1]) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' Let f ∈ C∞(S2, R3) ∩ Imm(S2, R3) be a regular surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' By [43, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content='33] the Gauss map of f is surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' Thus the image of qf is not contained in a single hyperplane of R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' Furthermore, � S2 qf(x)|qf(x)|dm = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' Thus, by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content='7, there exists a unique convex body (up to translation) with surface area measure given by µqf .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' By Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content='1 the surface f and the convex body are SRNF distance 0 from each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' The WFR metric as a diffeomorphic optimization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' In this section, we will generalize the results of the previous sections for the Wasserstein Fisher Rao distance on any manifold and for any coeffecient δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' Thus characterizing the Wasserstein Fisher Rao distance as a diffeomorphic optimization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' Let N be a smooth, connected, compact, oriented Riemannian manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' Define 10 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' BAUER, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' HARTMAN, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' KLASSEN the cone over N via C(N) := (N × R≥0)/(N × {0}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' If we let d denote the geodesic distance on N and fix some δ ∈ (0, ∞), then we can define a metric on C(N) via dC(N)((n1, r1), (n2, r2))2 = 4δ2r2 1 + 4δ2r2 2 − 8δ2r1r2cos(d(n1, n2)/2δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' Let M be another smooth, connected, compact, oriented Riemannian manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' Any function q : M → C(N) can be decomposed into component functions by q(x) = (q(x), q◦(x)) where q : M → N and q◦ : M → R≥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' We can thus define ˆq : M → R≥0 via for all x ∈ M, ˆq(x) = √ 2δq◦(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' Given q1, q2 : M → C(N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' The L2 distance between q1 and q2 is given by dL2(q1, q2)2 = � M dC(N)(q1(x), q2(x))2dm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' By decomposing q1 and q2, we can alternatively write (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content='1) dL2(q1, q2)2 = � M ˆq1(x)2dm + � M ˆq2(x)2dm − 2 � M ˆq1(x) ˆq2(x)cos(d(q1(x), q2(x))/2δ)dm The L2 cost of a function q : M → C(N) as the distance from q to the function that maps all of M to the cone point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' In particular, using the decomposition of q, this distance is given by dL2(0, q)2 = � M ˆq(x)2 dm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' Thus, the space of L2-functions from M to C(N) as L2(M, C(N)) := {q : M → C(N) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' dL2(0, q)2 < ∞} and we equip L2(M, C(N)) with the metric dL2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' We define the right action of the diffeomorphisms of on L2(M, C(N)) component-wise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' We treat ˆq as a half density and define the action of Γ(M) on this component as the action on half-densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' Thus, we define the action of Γ(M) on L2(M, C(N)) given by L2(M, C(N)) × Γ(M) → L2(M, C(N)) via (q, ˆq), γ �→ � q ◦ γ, ˆq ◦ γ · � |Dγ| � The main result of this section is to show that the Wasserstein Fisher Rao distance can written as the distance between the orbits associated with the measures: Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' Let N be a smooth connected compact Riemannian manifold and M be a smooth connected compact Riemannian manifold of dimension 2 or higher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=') For all µ1, µ2 ∈ M(N) and q1, q2 ∈ L2(M, C(N)) such that µ1 = q1∗νq1 and µ2 = q2∗νq2 we have WFRδ(µ1, µ2) = inf γ∈Γ(N) dL2(q1, q2 ∗ γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=') Moreover, for all µ ∈ M(N) there exists q ∈ L2(M, C(N)) such that µ = q∗νq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' If µ is a finitely supported measure given by µ = �n i=1 aiδui, then one can choose q piece wise constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' More specifically, the function q given by q(x) = �� uj, � aj area(σj) � if 1 ≤ j ≤ n (u1, 0) if n < j ≤ m , where {σj}m j=1 is a subdivision of the canonical triangulation of M with m ≥ n, satisfies µ = q∗νq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' 11 Before we are able to prove this theorem, we will show again several technical lemmas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' Therefore we will consider specific measures associated with functions q ∈ L2(M, C(N)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' First, we define νq ∈ M(M) such that for any U ⊆ M open νq(U) = � U ˆq(x)2dm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' Note that νq ≪ m and νq m = ˆq2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' Further, we can define a pushforward of νq via q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' In particular, for every q ∈ L2(M, C(N)), we can define a Borel measure on N given by µq := q∗νq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' In other words for all U ⊆ N open µq(U) = � q−1(U) ˆq2(x)dm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' Now we will show that the orbit of any q ∈ L2(M, C(N)) under the action of Γ(M) is mapped to the same measure on N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' Let q ∈ L2(M, C(N)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' Then for all γ ∈ Γ(M), µq = µq∗γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' Let U ⊆ N open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' Then µq∗γ(U) = � γ−1(q−1(U)) (ˆq ◦ γ(x) · � |Dγ|)2dm = � γ−1(q−1(U)) ˆq ◦ γ(x)2 · |Dγ|dm = � q−1(U) ˆq(x)2dm = µq(U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' Therefore, we can map each orbit of q ∈ L2(M, C(N)) under the half density action by Γ(M) to a measure on N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' As in the previous section, we will first show the result for piecewise constant functions and extend by continuity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' We prove the piecewise constant case in the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' Let d ≥ 2 and M be a smooth, connected, compact, oriented Riemannian d-dimensional manifold with or without boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' Given two piecewise constant functions q1, q2 : M → C(N), inf γ∈Γ(M) dL2(q2, q2 ∗ γ) = WFRδ(µq1, µq2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' Let {σi}m i=1 and {τj}n j=1 be triangulations of M such that q1 is constant on each σi and q2 is constant on each τj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' Let ˆq1 : M → R, q1 : M → N be the decomposition of q1 and ˆq2 : M → R, q2 : M → M be the decomposition of q2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' Define a function ⟨·, ·⟩ : N × N → R given via ⟨u, v⟩ = cos(d(u, v)/2δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' A brief computation shows inf γ∈Γ(M) d2 L2(q1, q2 ∗ γ) = m � i=1 ai + n � j=1 bj − 2 sup γ∈Γ(M) � M ˆq1(x) ˆq2(γ(x)) � |Dγ|⟨q1(x), q2(γ(x))⟩dm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' Let A be the set of all discrete semi-couplings from µq1 to µq2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' Recall WFRδ(µq1, µq2)2 = m � i=1 ai + n � j=1 bj − 2 sup (A,B)∈A m � i=1 n � j=1 � AijBij⟨ui, vj⟩ Therefore, the theorem is equivalent to showing sup (A,B)∈A m � i=1 n � j=1 � AijBij⟨ui, vj⟩ = sup γ∈Γ(S2) � M ˆq1(x) ˆq2(γ(x)) � |Dγ|⟨q1(x), q2(γ(x))⟩dm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' 12 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' BAUER, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' HARTMAN, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' KLASSEN Claim 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' Assume that (A, B) is a discrete semi-coupling from µq1 to µq2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' Then for all ǫ > 0 there is a PL homeomorphism γ : M → M such that ������ � M ˆq1(x) ˆq2(γ(x)) � |Dγ|⟨q1(x), q2(γ(x))⟩dm − � i,j � AijBij⟨ui, vj⟩ ������ < ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' Proof of Claim 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' Let (A, B) be a discrete semi-coupling from µq1 to µq2 such that for each 1 ≤ i ≤ m and 1 ≤ j ≤ n, Aij, Bij > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' We will first prove the claim for this restricted case and extend it to all semi-couplings by continuity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' First we choose a real number r ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' For each 1 ≤ i ≤ m, subdivide σi into n smaller d-simplexes σij such that ˆq1 2 = Aij/m(σij).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' Similarly, for each 1 ≤ j ≤ n, subdivide τj into m smaller d-simplexes τij such that ˆq2 2 = Bij/m(τij).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' For each 1 ≤ i ≤ m and 1 ≤ j ≤ n, choose a smaller d-simplex ˜σij, whose closure is contained in the interior of σij, such that m(˜σij) = rm(σij).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' Similarly, for each 1 ≤ i ≤ m and 1 ≤ j ≤ n, choose a smaller d-simplex ˜τij, whose closure is contained in the interior of τij, such that m(˜τij) = rm(τij).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' We now construct an orientation preserving PL homeomorphism γr : M → M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' First, for each 1 ≤ i ≤ m and 1 ≤ j ≤ n, define γr : ˜σij → ˜τij to be a PL orientation preserving homeomorphism with constant area multiplication factor, |Dγr| = m(τij)/m(σij).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' Note that M − \uf8eb \uf8ed m � i=1 n� j=1 ˜σo ij \uf8f6 \uf8f8 is homeomorphic to M − \uf8eb \uf8ed m � i=1 n � j=1 ˜τ o ij \uf8f6 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' Hence, we can extend the homeomorphism γr defined on the ˜σij’s to a homeomorphism from M to M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' Note that on each ˜σij, ˆq2 2(γr(x))|Dγr| = Bij/m(σij).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' Write M = M1 ∪ M2, where M1 = m� i=1 n� j=1 ˜σij and M2 = M − M1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' A simple computation shows � M1 ˆq1(x) ˆq2(γr(x)) � |Dγr|⟨q1(x), q2(γr(x))⟩dm = m � i=1 n � j=1 � ˜σij ˆq1(x) ˆq2(γr(x)) � |Dγr|⟨q1(x), q2(γr(x))⟩dm = m � i=1 n � j=1 � AijBij m(σij) m(˜σij)⟨ui, vj⟩ = m � i=1 n � j=1 � rAij � rBij⟨ui, vj⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' Meanwhile by the Schwarz inequality, ���� � M2 ˆq1(x) ˆq2(γr(x)) � |Dγr|⟨q1(x), q2(γr(x))⟩dm ���� ≤ � M2 ˆq1(x) ˆq2(γr(x)) � |Dγr|dm ≤ �� M2 ˆq1 2dm �� M2 ˆq2 2(γr(x))|Dγr|dm = � (1 − r) � M ˆq1 2dm � (1 − r) � M ˆq2 2dm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' So as we let r → 1, � M1 ˆq1(x) ˆq2(γr(x)) � |Dγr|⟨q1(x), q2(γr(x))⟩dm → m � i=1 n � j=1 � AijBij⟨ui, vj⟩ and � M2 ˆq1(x) ˆq2(γr(x)) � |Dγr|⟨q1(x), q2(γr(x))⟩dm → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' 13 Hence, � M ˆq1(x) ˆq2(γr(x)) � |Dγr|⟨q1(x), q2(γr(x))⟩dm → m � i=1 n � j=1 � AijBij⟨ui, vj⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' Thus Claim 2 follows for the case in which for each 1 ≤ i ≤ m and 1 ≤ j ≤ n, Aij > 0 and Bij > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' The general case then follows immediately from the continuity of m � i=1 n � j=1 � AijBij⟨ui, vj⟩ as a function of (A, B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' This completes the proof of Claim 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' It follows that sup γ∈Γ(S2) � M ˆq1(x) ˆq2(x)⟨q1(x), q2(x)⟩dm ≥ sup (A,B)∈A m � i=1 n � j=1 � AijBij⟨ui, vj⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' We are left to show the opposite inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' Claim 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' Assume γ is a PL-homeomorphism from M to M, then there exists a discrete semi- coupling (A, B) such that sup γ∈Γ(M) � M ˆq1(x) ˆq2(γ(x)) � |Dγ|⟨q1(x), q2(γ(x))⟩dm ≤ sup (A,B)∈A m � i=1 n � j=1 � AijBij⟨ui, vj⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' Proof of Claim 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' Let γ : M → M be an orientation preserving PL homeomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' For 1 ≤ i ≤ m and 1 ≤ j ≤ n, define σij = γ−1(τj) ∩ σi and define τij = γ(σij).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' Now define two (m + 1) × (n + 1) matrices A and B via: For 1 ≤ i ≤ m and 1 ≤ j ≤ n, Aij = � σij ˆq1 2dm and Bij = � τij ˆq2 2dm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' For 0 ≤ i ≤ m, B0i = 0 and Ai0 = ai − n � j=1 � σij ˆq1 2dm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' For 0 ≤ j ≤ n, Aj0 = 0 and B0j = bj − m � i=1 � τij ˆq2 2dm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' The pair of matrices (A, B) is a discrete semi-coupling from µq1 to µq2 by construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' We say that (A, B) is the semi-coupling corresponding to the homeomorphism γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' Denote the area multiplication factor of γ on σij by mij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' Then by the Schwarz inequality, � σij ˆq1(x) ˆq2(γ(x)) � |Dγ|⟨ui, vj⟩dm ≤ �� σij ˆq1 2(x)dm �� σij ˆq2 2(γ(x))|Dγ|dm⟨ui · vj⟩ = �� σij ˆq1 2(x)dm �� τij ˆq2 2(x)dm⟨ui · vj⟩ = � Aij � Bij⟨ui · vj⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' Summing over all i and j we obtain: � M ˆq1(x) ˆq2(γ(x)) � |Dγ|⟨q1(x), q2(γ(x))⟩dm = � i,j � σij ˆq1(x) ˆq2(γ(x)) � |Dγ|⟨q1(x), q2(γ(x))⟩dm ≤ � i,j � Aij � Bij⟨ui · vj⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' 14 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' BAUER, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' HARTMAN, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' KLASSEN This completes the proof of Claim 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' It follows that, sup γ∈Γ(M) � M ˆq1(x) ˆq2(γ(x)) � |Dγ|⟨q1(x), q2(γ(x))⟩dm ≤ sup (A,B)∈A m � i=1 n � j=1 � AijBij⟨ui · vj⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' and thus the lemma is proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' To extend the results to all of L2(M, C(N)) we will need the following continuity result: Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' The map (L2(M, C(N)), dL2) → (M(N), WFRδ) defined via q �→ q∗νq is Lipschitz continuous with Lipschitz constant K = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' Let q1, q2 ∈ L2(M, C(N)), µq1 = q1∗νq1, and µq2 = q2∗νq2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' For any semi-coupling (γ1, γ2) ∈ Γ(µq1, µq2), WFRδ(µq1, µq2) ≤ � Jδ(γ1, γ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' Thus, to prove the theorem we must construct (γ1, γ2) ∈ Γ(µq1, µq2) such that Jδ(γ1, γ2) = dL2(q1, q2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' To construct such a semi-coupling we first construct ρ : M → N×N defined a the first component maps of q1 and q2 on the first and second factor respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' the map is given by ρ(x) = (q1(x), q2(x)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' Since q1 and q2 are individually measurable, then so is ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' We can then define γ1, γ2 ∈ M(N × N) via γ1 = ρ∗νq1 and γ2 = ρ∗νq2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' Claim 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' The pair of measures, (γ1, γ2) is a semi-coupling from µq1 to µq2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' Proof of claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' Let U ⊆ N be open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' Thus, γ1(U × N) = νq1 � ρ−1(U × N) � = νq1 � q1−1(U) ∩ q2−1(N) � = νq1 � q1−1(U) � = µq1(U) and γ2(N × U) = νq2 � ρ−1(N × U) � = νq1 � q1−1(N) ∩ q2−1(U) � = νq1 � q2−1(U) � = µq2(U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' So (γ1, γ2) is a semi-coupling from µq1 to µq2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' Recall from the definition of the functional J we need to construct γ ∈ M(N ×N) such that γ1, γ2 ≪ γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' Define γ = ρ∗m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' We know µq1, µq2 ≪ m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' Thus, by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content='2, γ1, γ2 ≪ γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' Furthermore, ˆq1 2 = µq1 m = γ1 γ ◦ ρ a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' and ˆq2 2 = µq2 m = γ2 γ ◦ ρ a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' So,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' Jδ(γ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' γ2) =µ1(N) + µ2(N) − 2 � N×N √γ1γ2 γ (u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' v)cos(d(u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' v)/2δ)dγ(u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' v) = � N×N γ1 γ dγ + � N×N γ2 γ dγ − 2 � N×N �γ1 γ (u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' v)γ2 γ (u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' v)cos(d(u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' v)/2δ)dγ(u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' v) = � ρ−1(N×N) γ1 γ ◦ ρ dm + � ρ−1(N×N) γ2 γ ◦ ρ dm − 2 � ρ−1(N×N) �γ1 γ ◦ ρ(x)γ2 γ ◦ ρ(x)cos(d(ρ(x))/2δ)dm = � M ˆq1(x)2 dm + � M ˆq2(x)2 dm − 2 � M ˆq1(x) ˆq2(x)cos(d(q1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' q2)/2δ)dm = dL2(q1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' q2) Thus,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' WFRδ(µq1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' µq2) ≤ � Jδ(γ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' γ2) = 1 · dL2(µq1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' µq2) 15 Finally,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' we can leverage this continuity result to complete the proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' Proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' Let µ1, µ2 ∈ M(N) and q1, q2 ∈ L2(M, C(N)) such that µ1 = q1∗νq1 and µ2 = q2∗νq2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' By an argument analogous to the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content='1 we can conclude inf γ∈Γ(M) dL2(q1, q2 ∗ γ) = WFRδ(µ1, µ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' This concludes the the proof of part a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' Let µ = �n i=1 aiδui be a finitely supported measure on N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' By [48], M admits a canonical PL structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' Let m ≥ n and subdivide the triangulation of M into m simplices given by σj for 1 ≤ j ≤ m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' Let x ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' Thus, there exists 1 ≤ j ≤ m such that x ∈ σj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' Thus we define q(x) = �� uj, � aj area(σj) � if 1 ≤ j ≤ n (u1, 0) if n < j ≤ m .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' Let U ⊆ N, then µ(U) = � i|ui∈U ai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' Meanwhile, q−1(U) = � i|ui∈U σi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' Thus, � q−1(U) ˆq2(x)dm = � i|ui∈U � σi ai area(σi)dm = � i|ui∈U ai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' To complete the proof of part b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=') we will extend the result to the whole space by continuity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' For any µ ∈ M(N), let {µn} ⊆ M(N) be a sequence of finitely supported measures that converges to µ with respect to the Wasserstein Fisher Rao.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' In particular, {µn} is Cauchy with respect to WFRδ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' Note that for all n ∈ N,there exists a piecewise constant qn ∈ L2(M, C(N)) satisfying µn(U) = � qn−1(U) ˆqn(x)2dm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' Thus, we can construct a sequence of functions given by q∗ 0 = q0 an for all n ∈ N, q∗n+1 = qn+1 ∗ γn where γn is a PL homeomorphism from M to M such that dL2(q∗ n, qn+1 ∗ γn) = WFRδ(µn, µn+1) + 1 2n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' Note that the existence of such a γn is guaranteed by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' Since {µn} is Cauchy with respect to WFRδ, it follows that {q∗ n} is Cauchy with respect to dL2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' By completeness of (L2(M, C(N)), dL2), there exists a limit q ∈ L2(M, C(N)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' Let U ⊆ N open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' Thus, µ(U) = lim n→∞ µn(U) = lim n→∞ � qn−1(U) ˆqn(x)2dm = lim n→∞ � M ˆqn(x)2χqn−1(U)dm = � M lim n→∞ ˆqn(x)2χqn−1(U)dm = � M ˆq(x)2χq−1(U)dm = � q−1(U) ˆq(x)2dm Thus, µ = q∗νq This completes the proof of part b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=') of the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' REFERENCES [1] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' Alexandrov, Zur theorie der gemischten volumina von konvexen k¨orpern i, Mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' Sbornik NS, 1 (1938), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' 227– 251.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tAyT4oBgHgl3EQfcfeN/content/2301.00284v1.pdf'} +page_content=' [2] M.' metadata={'source': 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