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+Detection problems in the spiked matrix models
+Ji Hyung Jung∗, Hye Won Chung†, and Ji Oon Lee‡
+January 16, 2023
+Abstract
+We study the statistical decision process of detecting the low-rank signal from various signal-plus-
+noise type data matrices, known as the spiked random matrix models. We first show that the principal
+component analysis can be improved by entrywise pre-transforming the data matrix if the noise is non-
+Gaussian, generalizing the known results for the spiked random matrix models with rank-1 signals. As
+an intermediate step, we find out sharp phase transition thresholds for the extreme eigenvalues of spiked
+random matrices, which generalize the Baik-Ben Arous-P´ech´e (BBP) transition.
+We also prove the
+central limit theorem for the linear spectral statistics for the spiked random matrices and propose a
+hypothesis test based on it, which does not depend on the distribution of the signal or the noise. When
+the noise is non-Gaussian noise, the test can be improved with an entrywise transformation to the data
+matrix with additive noise. We also introduce an algorithm that estimates the rank of the signal when
+it is not known a priori.
+1
+Introduction
+One of the most natural approach to ‘signal-plus-noise’ type data is to consider spiked random matrices,
+which are the low-rank deformation of large random matrices. Most notable examples of spiked random
+matrices include spiked Wigner matrix and spiked Wishart matrix, where the signals are given as a low-rank
+mean matrix (spiked Wigner matrix) and a low-rank perturbation of the identity in its covariance matrix
+(spiked Wishart matrix). In this paper, we focus on the following three types of noisy data matrices, known
+as spiked random matrices, which generalize spiked Wigner/Wishart matrices:
+• Spiked Wigner matrix: the data matrix is of the form
+UΛ1/2U T + W,
+(1.1)
+where U = [u(1), u(2), . . . , u(k)] ∈ RN×k with U T U = Ik, and W is an N × N Wigner matrix. The
+signal-to-noise ratio (SNR) Λ = diag(λ1, λ2, . . . , λk) with λ1 ≥ λ2 ≥ . . . λk > 0 for some positive integer
+k, independent of N.
+∗Department of Mathematical Sciences, KAIST, Daejeon, 34141, Korea
+email: jhjung66@kaist.ac.kr
+†School of Electrical Engineering, KAIST, Daejeon, 34141, Korea
+email: hwchung@kaist.ac.kr
+‡Department of Mathematical Sciences, KAIST, Daejeon, 34141, and School of Mathematics, KIAS, Seoul, 02455, Korea
+email: jioon.lee@kaist.edu
+1
+arXiv:2301.05331v1  [math.ST]  12 Jan 2023
+
+• Rectangular matrix with spiked mean (additive model): the data matrix is of the form
+UΛ1/2V T + X,
+(1.2)
+where U = [u(1), u(2), . . . , u(k)] ∈ RM×k, V = [v(1), v(2), . . . , v(k)] ∈ RN×k with U T U = V T V =
+Ik, and X is an M × N random i.i.d. matrix whose entries are centered with variance N −1. The SNR
+Λ is given as in the spiked Wigner matrix.
+• Rectangular matrix with spiked covariance (multiplicative model): the data matrix is of the form
+(I + UΛU T )1/2X,
+(1.3)
+where U = [u(1), u(2), . . . , u(k)] with U T U = Ik and X is an M × N random i.i.d. matrix whose
+entries are centered with variance N −1. The SNR Λ is given as in the spiked Wigner matrix.
+Here, Ik is the identity matrix with rank k and we allow the case k = 0 where no signal is present. Throughout
+the paper, for the ease of notation, we denote by W an N × N Wigner matrix, and X an M × N random
+i.i.d. matrix.
+To describe the detection problems we consider in this paper, we first review the known results for the
+simplest case of the spiked random matrix models with rank-1 spike, i.e, k = 1 in (1.1), (1.2), and (1.3).
+Signal detection problem in rank-1 spiked random matrices: Many problems concerning the
+signal detection can be answered in the case with Gaussian noise and rank-1 spike. In this case, the spikes
+U = u and V = v are vectors, and SNR λ1 = λ, hence the spiked random matrices are of the following
+forms:
+√
+λuuT + W
+(1.4)
+√
+λuvT + X
+(1.5)
+(I + λuuT )1/2X.
+(1.6)
+For this case, reliable detection of the signal, i.e., detection with probability 1 − o(1) as M, N → ∞, is
+impossible if the SNR λ is below a certain threshold [44, 47]. The threshold is 1 as N → ∞ for spiked
+Wigner matrices; for spiked rectangular matrices, with additional assumption M/N → d0 as N → ∞, the
+threshold is √d0 for a general class of priors [50]. On the other hand, the signal can be reliably detected
+by the principal component analysis (PCA) if the SNR is above the threshold in which case the signal can
+actually be estimated [24, 41, 43].
+In the subcritical case where the signal is not reliably detectable, it is natural to consider a hypothesis
+test on the presence of the signal between H0 : λ = 0 and H1 : λ = ω, commonly referred to as the
+weak detection, which is also known as the sphericity test in the case the spike is drawn from the uniform
+distribution on the unit sphere, known as the spherical prior. As asserted by Neyman–Pearson lemma, the
+likelihood ratio (LR) test is optimal in the sense that it minimizes the sum of the Type-I error and the
+Type-II error. It was proved for several distributions of the spikes, called priors, that this sum for a spiked
+Wigner matrix converges to
+erfc
+�1
+4
+�
+− log (1 − λ)
+�
+(1.7)
+when H is Gaussian Orthogonal Ensemble (GOE), and for a spiked Wishart matrix
+erfc
+�
+1
+4
+�
+− log
+�
+1 − λ2
+d0
+��
+(1.8)
+2
+
+when XXT is a Wishart Ensemble; see, e.g., [47, 29, 28]. Here, erfc(·) is the complementary error function
+defined as
+erfc(x) =
+� ∞
+x
+e−t2dt.
+(1.9)
+Though optimal, the LR test is not efficient, and it is desirable to construct a test that does not depend
+on information about the prior, which is typically not known in many practical applications. In [22], an
+optimal and universal test for spiked Wigner matrices was proposed, which is based on the linear spectral
+statistics (LSS) of the data matrix, a linear functional defined as
+LN(f) =
+N
+�
+i=1
+f(µi)
+(1.10)
+for a given function f, where µ1, · · · µN are the eigenvalues of the data matrix. The test is extended to spiked
+rectangular matrices in [34], where the singular values of the data matrix is used instead of the eigenvalues.
+If the noise is non-Gaussian, it is possible to improve the PCA by transforming the data matrix entrywise
+for spiked Wigner matrices [42, 50] and for spiked rectangular matrices [34]. In this improved PCA, the
+threshold is lowered by a certain factor that depends on the Fisher information of the noise distribution.
+Below this threshold, the LSS-based test proposed in [22] for spiked Wigner matrices can also be improved by
+applying the entrywise transformation for the improved PCA. It is not known whether the reliable detection
+is impossible below the threshold except for the case of the spiked Wigner matrix with Rademacher prior
+[21].
+Spiked random matrices with general rank: The more relevant structure for application is that
+the latent signal contains multiple spikes, or a spike with a higher rank. For such models of spiked random
+matrices, similar to the cases with rank-1 spikes, it is natural to ask the following questions:
+• What is the spectral threshold for a reliable detection lower than the existing one for Gaussian noise
+if the noise is non-Gaussian?
+• Can we design an efficient algorithm to weakly detect the presence of signal (i.e., better than a random
+guess) when a reliable detection is not feasible?
+Contrary to the rank-1 spike case, the questions addressed above have never been answered, even for the
+simplest case of Gaussian noise. Furthermore, for the spikes with general rank, we need to consider another
+important problem of finding the rank of the spike in case it is not known a priori. While viable solutions to
+resolve the issue in the context of the community detection were suggested in [40, 16] for any spiked Wigner
+matrices and [49, 25] for spiked rectangular matrices, these methods are not applicable in the sub-critical
+case. To the best of our knowledge, there are no spectral algorithms for estimating the rank of signal in the
+sub-critical regime. We thus aim to the following question as well:
+• Can we design an efficient algorithm to estimate the rank of signal when a reliable detection is not
+feasible?
+Main contributions
+Our main contributions are mainly divided into three parts as follows:
+• (Strong detection) We prove that the PCA can be improved by an entrywise transformation if the
+noise is non-Gaussian, under a mild assumption on the distribution (prior) of the spike.
+3
+
+• (Weak detection I) We propose a universal test to detect the presence of signal with low computational
+complexity, based on the linear spectral statistics (LSS). The test does not require any prior information
+on the signal, and if the noise is Gaussian the error of the proposed test is optimal. For the spiked
+Wigner matrix or the additive model of the spiked rectangular matrix with the non-Gaussian noise,
+we suggest an improved test via an entrywise transformation.
+• (Weak detection II) We present an LSS-based test for estimating the rank of a signal when Λ = λI.
+Heuristically, it is possible to increase the SNR via an entrywise transformation. Here, we illustrate the
+main idea of the entrywise transformation for the spiked Wigner matrix of the form M = UΛ1/2U T + W.
+If |uiuT
+j |, |uivT
+j | ≪ Wij, then by applying a function q entrywise to
+√
+NY , we obtain a transformed matrix
+whose entries are
+q(
+√
+NMij) = q(
+√
+NWij +
+√
+NuiΛ1/2uT
+j ) ≈ q(
+√
+NWij) +
+√
+Nq′(
+√
+NWij)uiΛ1/2uT
+j ,
+where ui and vi denote i−th row vector of the signal matrix U and V , respectively. With negligible error, it
+is possible to approximate the coefficient q′(
+√
+NWij) in the second term in the right side by its expectation.
+(See Appendix B.3 for the proof) Then,
+q(
+√
+NMij) = q(
+√
+NWij +
+√
+NuiΛ1/2uT
+j ) ≈
+√
+N
+�
+q(
+√
+NWij)
+√
+N
++ E[q′(
+√
+NWij)]uiΛ1/2uT
+j
+�
+and the transformed matrix is approximately of the form U(Λ′)1/2U T + Q after a proper normalization,
+which becomes another spiked Wigner matrix with different SNR. By optimizing the transformation q, we
+find that the SNR is effectively increased (or equivalently, the threshold √d0 is lowered) in the PCA for the
+transformed matrix. The change of the threshold and a BBP-type transition for the largest eigenvalues of
+the transformed matrix can be rigorously proved; see Theorem 3.3 for a precise statement. We remark that
+the same idea works even if the SNR Λ is not a constant multiple of an identity matrix, and also a similar
+result holds for the additive model of spiked rectangular matrix (Theorem 3.4).
+For the multiplicative model of the form Y = (I + UΛU T )1/2X =: (I + UΓU T )X, with Λ = 2Γ + Γ2,
+the analysis is significantly more involved due to the following reason: Applying a function q entrywise to
+√
+NY , we find that
+q(
+√
+NYij) = q
+�√
+NXij +
+√
+N
+�
+ℓ
+uiΓuT
+ℓ Xℓj
+�
+≈ q(
+√
+NXij) +
+√
+Nq′(
+√
+NXij)
+�
+ℓ
+uiΓuT
+ℓ Xℓj
+≈
+√
+N
+�
+q(
+√
+NXij)
+√
+N
++ E[q′(
+√
+NXij)]
+�
+ℓ
+uiΓuT
+ℓ Xℓj
+�
+,
+and the transformed matrix is of the form UΓ′U T X +Q, which is not a spiked rectangular matrix anymore.
+Note that Q depends on X entrywise and thus it cannot be considered as an additive model, either.
+In Theorem 3.5, we prove the effective change of the SNR and the BBP-type transition for the multi-
+plicative model. The proof of Theorem 3.5 is based on a generalized version of the BBP transition that
+works with the matrix of the form UΓU T X +Q. We remark that the strategy for the proof, based on recent
+development of random matrix theory, can also be applied to prove a BBP-type transition for other models.
+As in the rank-1 case in [34], it is notable that the optimal entrywise transform for the multiplicative
+model is different from the one for the additive model. For the spiked Wigner matrix, the optimal transforms
+are given by −g′/g for the off-diagonal entries (and −g′
+d/gd for the diagonal entries), where g (and gd for
+the diagonal entries) is the density functions of them; the optimal transform for the additive model is also
+4
+
+given by −g′/g. However, for the multiplicative model, the optimal transform is a linear combination of
+the function −g′/g and the identity mapping. Heuristically, it is due to that the effective SNRs depend not
+only on Γ′ but also on the correlation between X and Q; the former is maximized when the transform is
+−g′/g while the latter is maximized when the transform is the identity mapping. We also remark that the
+effective SNRs after the optimal entrywise transform is larger in the additive model, which suggests that the
+detection problem is fundamentally harder for the multiplicative model.
+With the BBP-type transition for the largest eigenvalues of the transformed matrices, it is also possible
+to improve the performance of several statistical inferences [13, 35, 46]. One of the consequences is that the
+corresponding eigenspace is adjacent to its true spike U in the sense of direction of arrival (DoA) [23]. In
+other words, we can not only reliably estimate the number of spikes by parallel analysis (PA) [27], but also
+approximately recover the true spikes and the corresponding SNRs.
+For the subcritical case where it is impossible to reliably detect the signal by the improved PCA, we
+propose algorithms for weak detection, based on the central limit theorem (CLT) of the LSS, Theorems 5.2,
+5.3, 5.5, and 5.6, analogous to the ones introduced in [22]. More precisely, assuming the SNRs are uniform
+i.e., Λ = λI, we propose an algorithm for a hypothesis test between
+Hk1 : k = k1,
+Hk2 : k = k2
+(1.11)
+for non-negative integers k1 < k2. While it may seem obvious, it has not been even known in the simple
+case k1 = 0 whether the detection becomes easier as k2 increases. Our test in Algorithm 2 verifies the claim
+since the error of the proposed test is an increasing function of (k2 − k1) as in Theorem 4.2. As in [22], the
+proposed tests are universal, and the various quantities in it can be estimated from the observed data. The
+test can further be improved by applying the same entrywise transformation we used for the PCA (Algorithm
+3) if the data matrix is of additive type (spiked Wigner matrix or rectangular matrix with spiked mean),
+and it also can be adapted to the rank detection problem where we need to estimate the rank k of the signal
+without knowing the candidates k1 and k2 a priori (Algorithm 4).
+The main mathematical achievement of the second part is the CLT for the LSS of spiked random matrices
+with general ranks.
+For a rank-1 spiked Wigner matrix, the CLT was first proved for a special spike
+1
+√
+N (1, 1, . . . , 1)T in [9] and later extended for a general rank-1 spike by comparison with the special case
+[22]. However, the proof in [9] is not readily extended to the spiked Wigner matrices with higher ranks and the
+spiked rectangular matrices. In this paper, we overcome the difficulty by introducing a direct interpolation
+between the spiked random matrix and the corresponding pure noise matrix and tracking the change of the
+LSS. Furthermore, we will prove that the proposed entrywise transformation for the data matrix of additive
+type also effectively changes the SNR, and that the LSS of the transformed matrix is also asymptotically
+Gaussian; this result was proved previously only for rank-1 spiked Wigner matrices in [22]. Thus, the error
+from the proposed test decreases after the transformation as for spiked Wigner matrices in [22].
+Related works
+Spiked random matrix models were first introduced by Johnstone [31]. The model can be applied to various
+problems such as community detection [1] and submatrix localization [19]. The transition of the largest
+eigenvalue was proved by Baik, Ben Arous, and P´ech´e [7] for spiked complex Wishart matrices and generalized
+by Benaych-Georges and Nadakuditi [14, 15]. For more results from random matrix theory about the extreme
+eigenvalues and the corresponding eigenvectors of spiked random matrices, we refer to [18] and references
+therein.
+The improved PCA based on the entrywise transformation was considered for rank-1 spiked Wigner
+matrices in [42, 50], where the transformation is chosen to maximize the effective SNR of the transformed
+5
+
+matrix. Detection problems for rank-1 spiked Wigner matrices were also considered, where the analysis is
+typically easier due to its symmetry and canonical connection with spin glass models. For more results on
+the rank-1 spiked Wigner matrices, we refer to [44, 50, 29, 22] and references therein.
+The testing problem for rank-1 spiked Wishart matrices with the spherical prior was considered by
+Onatski, Moreira, and Hallin [47, 48], where they proved the optimal error of the hypothesis test. It is later
+extended to the case where the entries of the spikes are i.i.d. with bounded support (i.i.d. prior) by El
+Alaoui and Jordan [28]. See also [32, 44, 24, 41, 43, 12] for more about detection limits in statistical learning
+theory.
+Models with sparse or generative structure of the spike have extensively studied in the past literature.
+Various statistical and algorithmic methods are applicable to the case where SNR is smaller than the spectral
+threshold. In particular, it can be seen that the sparsity of the spikes and the dimension of the latent vector
+constituting the generative spike prior actually serve to lower the threshold for the SNR to which several
+algorithms are applicable; see [5, 20] and references therein.
+Organization of the paper
+The rest of the paper is organized as follows: In Section 2, we introduce the precise definitions of models
+and relevant previous consequences. In Section 3, we state our results on the improved PCA. In Section 4,
+we propose algorithms for LSS-based tests and a test for rank estimation, and analyze their performance.
+In Section 5, we state general results on the CLT for the LSS. We conclude the paper in Section 6 with
+the summary of our works and future research directions. In Appendix A, we consider examples of spiked
+random matrices and provide results from numerical experiments.
+In Appendices B and C, we provide
+technical details of the proofs.
+2
+Preliminaries
+In this section, we introduce the precise definition of the models and previous results for the spiked random
+matrices.
+2.1
+Definitions of models
+The noise matrices are defined as follows:
+Definition 2.1 (Wigner matrix). An N ×N symmetric matrix W = (Wij) is a (real) Wigner matrix if Wij
+(i, j = 1, 2, . . . , N) are independent real random variables such that
+• For all i < j, NE[W 2
+ij] = 1, N
+3
+2 E[W 3
+ij] = w3, and N 2E[W 4
+ij] = w4 for some w3, w4 ∈ R.
+• For all i, NE[W 2
+ii] = w2 for some constant w2 ≥ 0.
+• For any positive integer p, there exists Cp, independent of N, such that N
+p
+2 E[W p
+ij] ≤ Cp for all i ≤ j.
+Definition 2.2 (Random rectangular matrix). An M × N matrix X = (Xij) is a (real) random rectangular
+matrix if Xij (1 ≤ i ≤ M, 1 ≤ j ≤ N) are independent real random variables such that
+• For all i, j, E[Xij] = 0, NE[X2
+ij] = 1, N
+3
+2 E[X3
+ij] = w3, and N 2E[X4
+ij] = w4 for some constants w3, w4.
+• For any positive integer p, there exists Cp, independent of N, such that N
+p
+2 E[Xp
+ij] ≤ Cp for all i, j.
+The spiked random matrices are defined as follows:
+6
+
+Definition 2.3 (Spiked Wigner matrix). An N × N matrix M = UΛ1/2U T + W is a spiked Wigner matrix
+with the SNR (matrix) Λ if W is a Wigner matrix and the spike U = [u(1), u(2), . . . , u(k)] ∈ RN×k with
+U T U = Ik.
+Definition 2.4 (Spiked rectangular matrix - additive model). An M ×N random matrix Y = UΛ1/2V T +X
+is a rectangular matrix with spiked mean U, V and the SNR (matrix) Λ if X is a random rectangular matrix
+and the spikes U = [u(1), u(2), . . . , u(k)] ∈ RM×k, V = [v(1), v(2), . . . , v(k)] ∈ RN×k with U T U = V T V =
+Ik.
+Definition 2.5 (Spiked rectangular matrix - multiplicative model). An M × N random matrix Y = (I +
+UΛU T )1/2X is a rectangular matrix with spiked covariance U and the SNR (matrix) Λ if X is a rectangular
+matrix and U = [u(1), u(2), . . . , u(k)] ∈ RM×k with U T U = Ik.
+We assume throughout the paper that the SNR matrix Λ is a k × k diagonal matrices with Λii = λi and
+λ1 ≥ λ2 ≥ . . . λk ≥ 0, and M
+N → d0 ∈ (0, ∞) as M, N → ∞.
+2.2
+Principal component analysis
+Here are the results for principal components of spiked models in the context of random matrix theory.
+Spiked Wigner matrix
+Let M be the spiked Wigner matrix.
+The empirical spectral measure of M converges to the Wigner’s
+semicircle law µsc, i.e., if we denote by µ1 ≥ µ2 ≥ · · · ≥ µN the eigenvalues of M, then
+1
+N
+N
+�
+i=1
+δµi(x)dx → dµsc(x)
+(2.1)
+weakly in probability as N → ∞, where
+dµsc(x) =
+√
+4 − x2
+2π
+1(−2,2)(x)dx.
+(2.2)
+The k largest eigenvalue has the following (almost sure) limit: for 1 ≤ i ≤ k
+• If λi > 1, then µi → √λi +
+1
+√λi .
+• If λi < 1, then µi → 2.
+Sample covariance matrix
+Let S = Y Y T be the sample covariance matrix (Gram matrix) derived from a spiked rectangular matrix
+Y . The empirical spectral measure of S converges to the Marchenko–Pastur law µMP , i.e., if we denote by
+µ1 ≥ µ2 ≥ · · · ≥ µM the eigenvalues of S, then
+1
+M
+M
+�
+i=1
+δµi(x)dx → dµMP (x)
+(2.3)
+weakly in probability as M, N → ∞, where for M ≤ N
+dµMP (x) =
+�
+(x − d−)(d+ − x)
+2πd0x
+1(d−,d+)(x)dx,
+(2.4)
+7
+
+with d± = (1 ± √d0)2. The k largest eigenvalue has the following (almost sure) limit: for 1 ≤ i ≤ k
+• If λi > √d0, then µi → (1 + λi)(1 + d0
+λi ).
+• If λi < √d0, then µi → d+ = (1 + √d0)2.
+This in particular shows that the detection can be reliably done by PCA if λ > √d0. We remark that the
+results above hold for both the additive model and the multiplicative model.
+2.3
+Linear spectral statistics
+We introduce the central limit theorems for null models.
+Spiked Wigner matrix
+The proof of the Gaussian convergence of the LR in [8, 10] is based on the recent study of linear spectral
+statistics, defined as
+LY (f) =
+N
+�
+i=1
+f(µi)
+(2.5)
+for a function f, where µ1 ≥ µ2 ≥ . . . µN are the eigenvalues of M. As the Wigner’s semicircle law in (2.1)
+suggests, it is required to consider the fluctuation of the LSS about
+N
+� 2
+−2
+f(x) dµsc(x).
+The CLT for the LSS is the statement
+�
+LM(f) − N
+� 2
+−2
+f(x) dµsc(x)
+�
+⇒ N(mM(f), VM(f)),
+(2.6)
+where the right-hand side is the Gaussian random variable with the mean mM(f) and the variance VM(f).
+The CLT was proved for the null case (λ = 0). We will show that the CLT also holds under the alternative
+and the mean mM(f) depends on λ while the variance VM(f) does not.
+Spiked rectangular matrices
+The LSS for the spiked rectangular matrices defined as
+LY (f) =
+M
+�
+i=1
+f(µi)
+(2.7)
+for a function f, where µ1 ≥ µ2 ≥ . . . µM are the eigenvalues of S = Y Y T . As the Marchenko–Pastur law in
+(2.3) suggests, it is required to consider the fluctuation of the LSS about
+M
+� d+
+d−
+f(x) dµMP (x).
+The CLT for the LSS is the statement
+�
+LY (f) − M
+� d+
+d−
+f(x) dµMP (x)
+�
+⇒ N(mY (f), VY (f)),
+(2.8)
+8
+
+where the right-hand side is the Gaussian random variable with the mean mY (f) and the variance VY (f).
+The CLT was proved for the null case (λ = 0). We will show that the CLT also holds under the alternative
+and the mean mY (f) depends on λ while the variance VY (f) does not.
+3
+Main result I - Improved PCA
+In this section, we state our first main results on the improvement of PCA by entrywise transformations and
+provide the results from numerical experiments.
+3.1
+Improved PCA
+We introduce the following assumptions for the spike and the noise.
+Assumption 3.1. For the spike U (and also V in the additive model), we assume, for φ ≤ 1/2,
+1. the spikes are φ-localized with high probability, i.e. ∥U∥∞, ∥V ∥∞ ≺ N −φ
+2. the spike matrix is φ-orthonormal with high probability, i.e. ∥U T U − Ik∥F , ∥V T V − Ik∥F ≺ N −φ,
+and so the spikes are sampled from Stiefel manifold of orthonormal k-frames in RM or RN with high
+probability.
+For the noise, let P be the distribution of the normalized entries
+√
+NWij(i ̸= j) in 2.1 and
+√
+NXij in
+2.2. Further, for the spiked Wigner matrices, let Pd be the distribution of the normalized diagonal entries
+√
+NWii in 2.1. We assume the following:
+1. The density functions g and gd of P and Pd, respectively, are smooth, positive everywhere, and sym-
+metric (about 0).
+2. For any fixed (N-independent) D, the D-th moments of P and Pd are finite.
+3. The functions h = −g′/g, hd = −g′
+d/gd and their all derivatives are polynomially bounded in the sense
+that |h(ℓ)(w)|, |h(ℓ)
+d (w)| ≤ Cℓ|w|Cℓ for some constant Cℓ depending only on ℓ.
+The first condition on the prior implies that the spike is not necessarily delocalized, i.e., some entries of
+the signal can be significantly larger than N −1/2. The key examples of the prior are as follows:
+Example 3.2. We can consider the following examples of the spike prior:
+1. the spherical prior, where u(ℓ) (and v(ℓ)) are i.i.d. drawn uniformly from the unit sphere, or
+2. the i.i.d. prior, where the entries u1(ℓ), . . . , uM(ℓ) (respectively, v1(ℓ), . . . , vN(ℓ)) are i.i.d. random
+variables from the probability measures µℓ (respectively, νℓ) with mean zero and variance M −1 (respec-
+tively N −1) such that for any integer p > 2
+E|ui(ℓ)|p, E|vj(ℓ)|p ≤
+Cp
+M 1+(p−2)φ
+for some (N-independent) constants Cp > 0 and φ ≤ 1
+2, uniformly on i, j and ℓ.
+We remark that for the spike Wigner matrices, due to normalization, the variance of the i.i.d. prior µℓ for
+ui(ℓ) is N −1.
+9
+
+Spiked Wigner matrix
+Given a spiked Wigner matrix M, we consider a family of the entrywise transformations
+hα(x) = −g′(x)
+g(x) + αx,
+hd(x) = −g′
+d(x)/gd(x)
+(3.1)
+for α ∈ R. We also consider the transformed matrix �
+M whose entries are
+�
+Mij =
+1
+�
+FgN h0(
+√
+NMij)(i ̸= j),
+�
+Mii =
+�
+w2
+Fg,dN hd
+��
+N
+w2
+Mii
+�
+,
+(3.2)
+where the Fisher information Fg and Fg,d of g and gd are given by
+Fg =
+� ∞
+−∞
+(g′(x))2
+g(x)
+dx,
+Fg,d =
+� ∞
+−∞
+g′
+d(x)2
+gd(x) dx.
+Note that Fg ≥ 1 where the equality holds only if g is the standard Gaussian.
+Then following theorem asserts that the effective SNRs of the transformed matrix for PCA are λℓFg,
+which generalizes Theorem 4.8 in [50].
+Theorem 3.3. Let M be a spiked Wigner matrix in Definition 2.3 satisfying Assumption 3.1 with φ > 1/4.
+Let �
+M be the transformed matrix obtained as in (3.2) and (�µℓ, �u(ℓ)) the pair of ℓ-th largest eigenvalue and
+the corresponding eigenvector of �
+M. Then, almost surely, for 1 ≤ ℓ ≤ k
+• If λℓ >
+1
+Fg , then �µℓ →
+�
+λℓFg +
+1
+√
+λℓFg and |�u(ℓ)T u(ℓ)|2 → 1 −
+1
+λℓFg ,
+• If λℓ <
+1
+Fg , then �µℓ → 2 and |�u(ℓ)T u(ℓ)|2 → 0.
+For the proof, we adapt the strategy in [50], where the key observation is that the transformed matrix is
+approximately equal to another spiked Winger matrix. See Appendix B.2 for the detail of the proof.
+We remark that h0 is the optimal (up to constant factor) among all entrywise transformations. See
+Appendix B.5.1 for the proof of it.
+Spiked rectangular matrices
+For a spiked rectangular matrix Y , we consider the family of the entrywise transformations hα(x) defined in
+(3.1) and transformed matrices �Y (α) whose entries are
+�Y (α)
+ij
+=
+1
+�
+(α2 + 2α + Fg)N
+hα(
+√
+NYij).
+(3.3)
+Note that
+For the additive model, we again show that the effective SNRs of the transformed matrix for PCA are
+{λℓFg}ℓ.
+Theorem 3.4. Let Y be a spiked rectangular matrix in Definition 2.4 satisfying Assumption 3.1 with φ >
+1/4. Let �Y ≡ �Y (0) be the transformed matrix obtained as in (3.3) with α = 0 and (�µℓ, �u(ℓ)) the pair of ℓ-th
+largest eigenvalue and the corresponding eigenvector of �Y �Y T . Then, almost surely, for 1 ≤ ℓ ≤ k
+• If λℓ >
+√d0
+Fg , then �µℓ → (1 + λℓFg)(1 +
+d0
+λℓFg ) and |�u(ℓ)T u(ℓ)|2 → 1 −
+d0(1+λℓFg)
+λℓFg(λℓFg+d0).
+• If λℓ <
+√d0
+Fg , then �µℓ → d+ = (1 + √d0)2 and |�u(ℓ)T u(ℓ)|2 → 0.
+10
+
+From Theorem 3.4, if λℓ >
+√d0
+Fg , we immediately see that the signal in the additive model can be reliably
+detected by the transformed PCA. Thus, the detection threshold in the PCA is lowered when the noise is
+non-Gaussian. We also remark that h0 is the optimal entrywise transformation (up to constant factor) as in
+the Wigner case; see Appendix B.5.2.
+For the proof, we adapt the strategy in [34], where the key observation is again that the transformed
+matrix is approximately equal to another spiked rectangular matrix. See Appendix B.3 for the detail of the
+proof.
+For the multiplicative model, we have the following result.
+Theorem 3.5. Let Y be a spiked rectangular matrix in Definition 2.5 satisfying Assumption 3.1 with φ >
+1/4. Let �Y ≡ �Y (αg,ℓ) be the transformed matrix obtained as in (3.3) with
+αg,ℓ :=
+−γℓFg +
+�
+4Fg + 4γℓFg + γ2
+ℓ F 2g
+2(1 + γℓ)
+and (�µℓ, �u(ℓ)) the pair of ℓ-th largest eigenvalue and the corresponding eigenvector of �Y �Y T . Then, almost
+surely,
+• If (λg)ℓ > √d0, then �µℓ → (1 + (λg)ℓ)(1 +
+d0
+(λg)ℓ ) and
+|�u(ℓ)T u(ℓ)|2 → 1 −
+(λg)ℓ + d0
+(λg)ℓ · ((λg)ℓ + 1),
+• If (λg)ℓ < √d0, then �µℓ → d+ = (1 + √d0)2 and |�u(ℓ)T u(ℓ)|2 → 0.
+where
+(λg)ℓ := γℓ + γ2
+ℓ Fg
+2
++
+γℓ
+�
+4Fg + 4γℓFg + γ2
+ℓ F 2g
+2
+.
+Note that
+(λg)ℓ ≥ γℓ + γ2
+ℓ Fg
+2
++
+γℓ
+�
+4 + 4γℓFg + γ2
+ℓ F 2g
+2
+= 2γℓ + γ2
+ℓ Fg ≥ 2γℓ + γ2
+ℓ = λℓ,
+and the inequality is strict if Fg > 1, i.e., g is not Gaussian.
+Note that unlike the additive model, we cannot determine αg without prior knowledge on the SNR.
+Nevertheless, we can apply the transformation h√
+Fg or h0, which effectively increase all SNRs simultaneously;
+see Appendix B.5.
+From Theorem 3.5, if (λg)ℓ > √d0, the signal can be reliably detected by the transformed PCA and the
+detection threshold in the PCA is lowered if the noise is non-Gaussian. We also remark that hαg,ℓ is the
+optimal entrywise transformation (up to constant factor) for the ℓ-th largest eigenvalue; see Appendix B.5.
+We finish this section with an outline of the proof of Theorem 3.5. We begin by justifying that the
+transformed matrix �Y is approximately of the form (Q+U�Γ
+1
+2 U T X), where �Γ = diag(�γ1, · · · , �γk). Then, the
+largest eigenvalue of �Y �Y T can be approximated by the largest eigenvalue of (Q+U�Γ
+1
+2 U T X)T (Q+U�Γ
+1
+2 U T X)
+for which we consider an identity
+(Q + U�Γ
+1
+2 U T X)T (Q + U�Γ
+1
+2 U T X) − zI = (QT Q − zI)(I + L(z)),
+11
+
+where
+L(z) = G(z)(XT U�Γ
+1
+2 U T Q + QT U�Γ
+1
+2 U T X + XT U�ΓU T X),
+G(z) = (QT Q − zI)−1.
+If z is an eigenvalue of (Q + U�Γ
+1
+2 U T X)T (Q + U�Γ
+1
+2 U T X) but not of QT Q, the determinant of (I + L(z))
+must be 0 and hence −1 is an eigenvalue of L(z). Since the rank of L(z) is at most 2k, we can find that the
+eigenvector of L(z) is a linear combination of vectors G(z)QT u(ℓ) and G(z)XT u(ℓ). Further, by using the
+facts in Example 3.2, we can observe that a linear combination of vectors G(z)QT u(ℓ) and G(z)XT u(ℓ) be
+a possible candidate for the ℓ-th eigenvector of L(z), and so of �Y T �Y i.e., for some aℓ, bℓ,
+L(z)(aℓG(z)QT u(ℓ) + bℓG(z)XT u(ℓ)) = −(aℓG(z)QT u(ℓ) + bℓG(z)XT u(ℓ)).
+(3.4)
+From the definition of L(z),
+L(z) · G(z)XT U = G(z)XT U�Γ
+1
+2 (U T QG(z)XT U) + G(z)QT U�Γ
+1
+2 (U T XG(z)XT U)
++ G(z)XT U�Γ(U T XG(z)XT U),
+and a similar equation holds for L(z) · G(z)QT U. It suggests that if U T QG(z)XT U and U T XG(z)XT U are
+concentrated around diagonal matrices where the entries are deterministic functions of z, then the left side
+of (3.4) can be well-approximated by a (deterministic) linear combination of G(z)QT u(ℓ) and G(z)XT u(ℓ).
+We can then find the location of the largest eigenvalue in terms of a deterministic function of z and conclude
+the proof by optimizing the function q.
+The concentration of random matrices U T QG(z)XT U and U T XG(z)XT U is the biggest technical chal-
+lenge in the proof, mainly due to the dependence between the matrices Q and X. We prove it by applying
+the technique of linearization in conjunction with resolvent identities and also several recent results from
+random matrix theory, most notably the local Marchenko–Pastur law.
+Once we find out the coefficients aℓ and bℓ in (3.4), the eigenvector localization is an easy corollary
+since the vector aℓG(z)QT u(ℓ) + bℓG(z)XT u(ℓ) must be a right singular vector of �Y with the corresponding
+singular value
+�
+(1 + (λg)ℓ)(1 +
+d0
+(λg)ℓ ). In this paper, we will not go into further detail on this part.
+The detailed proof of Theorem 3.5 can be found in Appendix B.4.
+4
+Main Result II - Weak Detection
+4.1
+Signal detection in rank-1 spiked models
+We begin by recalling the LSS-based detection algorithms for rank-1 spiked rectangular matrices in [34].
+Suppose that our goal is to detect the presence of the signal by the hypothesis test between H0 : λ = 0 and
+H1 : λ = ω where the SNR ω for the alternative hypothesis H1 is known. The key observation is that the
+variances of the limiting Gaussian distributions of the LSS in (2.7) do not depend on the SNR while the means
+do. If we denote by VY (f) the common variance, and mY (f)|H0 and mY (f)|H1 the means, respectively, our
+goal is to find a function that maximizes the relative difference between the limiting distributions of the LSS
+under H0 and under H1, i.e.,
+�����
+mY (f)|H1 − mY (f)|H0
+�
+VY (f)
+����� .
+(4.1)
+12
+
+Algorithm 1 Hypothesis test for a rank-1 spiked rectangular matrix
+Input: data Yij, parameters w4, ω
+Lω ← test statistic in (4.3)
+mω ← critical value in (4.7)
+if Lω ≤ mω then
+Accept H0
+else
+Reject H0
+end if
+As we will see in Theorem 5.5, the optimal function f is of the form C1φω + C2 for some constants C1 and
+C2, where
+φω(x) = ω
+d0
+�
+2
+w4 − 1 − 1
+�
+x − log
+��
+1 + d0
+ω
+�
+(1 + ω) − x
+�
+.
+(4.2)
+The test statistic we use is thus defined as
+Lω =
+M
+�
+i=1
+φω(µi) − M
+� d+
+d−
+φω(x) dµMP (x)
+= − log det
+��
+1 + d0
+ω
+�
+(1 + ω)I − Y Y T
+�
++ ω
+d0
+�
+2
+w4 − 1 − 1
+�
+(Tr Y Y T − M)
++ M
+� ω
+d0
+− log
+� ω
+d0
+�
+− 1 − d0
+d0
+log(1 + ω)
+�
+.
+(4.3)
+Theorem 8 in [34] asserts that Lω converges to a Gaussian,
+Lω ⇒ N(m(λ), V0).
+(4.4)
+Here, the mean of the limiting Gaussian distribution is given by
+m(λ) = −1
+2 log
+�
+1 − ω2
+d0
+�
++ ω2
+2d0
+(w4 ��� 3) − log
+�
+1 − λ2
+d0
+�
++ λ2
+d0
+�
+2
+w4 − 1 − 1
+�
+(4.5)
+with λ = 0 under H0 and λ = ω under H1, and the variance
+V0 = −2 log
+�
+1 − ω2
+d0
+�
++ 2ω2
+d0
+�
+2
+w4 − 1 − 1
+�
+.
+(4.6)
+Based on the asymptotic normality of Lω, we can construct a test in which we compute the test statistic
+Lω and compare it with the average of m(0) and m(ω), i.e.,
+mω := m(0) + m(ω)
+2
+= − log
+�
+1 − ω2
+d0
+�
++ ω2
+2d0
+�
+2
+w4 − 1 + w4 − 4
+�
+.
+(4.7)
+See Algorithm 1 for the detail.
+The limiting error of the proposed test, Algorithm 1, is given by
+err(ω) = P(Lω > mω|H0) + P(Lω ≤ mω|H1) → erfc
+�√V0
+4
+√
+2
+�
+,
+(4.8)
+13
+
+where V0 is the variance in (4.6) and erfc(·) is the complementary error function. If the noise X is Gaussian,
+w4 = 3 and the limiting error in (4.8) is
+erfc
+�√V0
+4
+√
+2
+�
+= erfc
+�
+1
+4
+�
+− log
+�
+1 − ω2
+d0
+��
+,
+and it coincides with the error of the LR test; see Section 2.2 of [34]. It shows that our test is optimal with
+the Gaussian noise.
+4.2
+Signal detection in rank-k spiked models
+When the rank of the spike is larger than 1, we first consider a simple case where the data is given as a
+spiked Wigner matrix and our goal is to construct an LSS-based algorithm for a hypothesis test between
+H0 : Λ = 0 and Hk : Λ = ωIk, where the rank k of the spike for the alternative hypothesis is known. Our
+starting point is the following test statistic, which was considered for the rank-1 spiked Wigner matrix in
+[22]:
+Lω = − log det
+�
+(1 + ω)I − √ωM
+�
++ ωN
+2
++ √ω
+� 2
+w2
+− 1
+�
+Tr M + ω
+�
+1
+w4 − 1 − 1
+2
+�
+(Tr M 2 − N).
+(4.9)
+If there is no signal present, Lω ⇒ N(m0, V0), where
+m0 = −1
+2 log(1 − ω) +
+�w2 − 1
+w4 − 1 − 1
+2
+�
+ω + (w4 − 3)ω2
+4
+,
+(4.10)
+V0 = −2 log(1 − ω) +
+� 4
+w2
+− 2
+�
+ω +
+�
+2
+w4 − 1 − 1
+�
+ω2.
+(4.11)
+For a rank-k spiked Wigner matrix, we can consider the same Lω as in (4.9) and prove that it also
+converges to a Gaussian with the same variance V0 but an altered mean mk. The following is the precise
+statement for the limiting distribution of Lω.
+Theorem 4.1. Let M be a rank-k spiked Wigner matrix with a spike U as in Definition 2.3 with Λ = ωIk
+for some nonnegative integer k. Then,
+Lω ⇒ N(mk, V0) ,
+(4.12)
+where the variance V0 is as in (4.11) and the mean mk is given by
+mk = m0 + k
+�
+− log(1 − ω) +
+� 2
+w2
+− 1
+�
+ω +
+�
+1
+w4 − 1 − 1
+2
+�
+ω2
+�
+= m0 + kV0
+2 .
+(4.13)
+Proof. Theorem 4.1 directly follows from Theorem 5.2 in Section 5.
+Since the mean of Lω depends on the rank of the spike, we can construct a hypothesis test between Hk1
+and Hk2 in (1.11) based on Theorems 4.1 and 4.4. In this test, for a given spiked Wigner matrix M, we
+compute Lω and compare it with the critical value m(k1+k2)/2,
+m(k1+k2)/2 := mk1 + mk2
+2
+.
+(4.14)
+14
+
+Algorithm 2 Hypothesis test for a spiked Wigner matrix
+Data: Mij, parameters w2, w4, λ
+Lω ← test statistic in (4.9),
+m(k1+k2)/2 ← critical value in (4.14) with (4.13)
+if Lω ≤ m(k1+k2)/2 then
+Accept H1
+else
+Accept H2
+end if
+See Algorithm 2 for the detail.
+In Theorems 5.2 and 5.5, we prove that the proposed test in Algorithm 2 is optimal among all CLT-based
+tests, in the sense that the error is minimized with the test statistic Lω also for spiked random matrices.
+Theorem 4.2. The error of the test, err(ω) = P(Lω > mω|H0)+P(Lω ≤ mω|H1), in algorithm 2 converges
+to
+erfc
+�
+k2 − k1
+4
+�
+V0
+2
+�
+.
+Proof. Theorem 4.2 is a direct consequence of Theorems 4.1 and 4.4. (See also Section 3 of [29] and the
+proof of Theorem 2 of [22].)
+Remark 4.3. When w4 = 3, we find that the error err(ω) converges to
+erfc
+�
+k2 − k1
+4
+�
+− log(1 − ω) +
+� 2
+w2
+− 1
+�
+ω
+�
+.
+(4.15)
+The optimal error for the weak detection, achieved by the LR test, coincides with the limiting error in (4.15)
+when the noise is Gaussian and the SNR ω is sufficiently small; see [33]. Thus, our proposed test is optimal
+in this case.
+The test in Algorithm 2 can be readily extended to the spiked rectangular matrices by replacing the
+test statistic in (4.9) with the following one, which was introduced in [34] for the rank-1 spiked rectangular
+matrices.
+Lω = − log det
+��
+1 + d0
+ω
+�
+(1 + ω)I − Y Y T
+�
++ ω
+d0
+�
+2
+w4 − 1 − 1
+�
+(Tr Y Y T − M)
++ M
+� ω
+d0
+− log
+� ω
+d0
+�
+− 1 − d0
+d0
+log(1 + ω)
+�
+.
+(4.16)
+We have the following results for the asymptotic normality of Gaussian fluctuation of Lω:
+Theorem 4.4. Let Y be a spiked rectangular matrix in Definition 2.4 or 2.5 with Λ = ωIk for some
+nonnegative integer k and λ ∈ (0, √d0) and w4 > 1. Then, for any spikes with U T U = V T V = Ik,
+Lω ⇒ N(mk, V0),
+(4.17)
+where the mean and the variance are given by
+mk = m0 + k
+�
+− log
+�
+1 − ω2
+d0
+�
++ ω2
+d0
+�
+2
+w4 − 1 − 1
+��
+(4.18)
+15
+
+and
+V0 = −2 log
+�
+1 − ω2
+d0
+�
++ 2ω2
+d0
+�
+2
+w4 − 1 − 1
+�
+(4.19)
+where
+m0 = −1
+2 log
+�
+1 − ω2
+d0
+�
++ ω2
+2d0
+(w4 − 3).
+(4.20)
+Theorem 4.4 directly follows from the general CLT result in Theorems 5.5. See Appendix C.4 for the
+detailed computation for the mean and the variance.
+With Theorem 4.4, we find that Algorithm 2 is available for the weak detection of the signal in the spiked
+rectangular matrices with the following change:
+• Data matrix is Yij (instead of Mij).
+• Test statistic Lω is defined by (4.16) (instead of (4.9)).
+• Critical value m(k1+k2)/2 is obtained by (4.14) with (4.20) (instead of (4.13)).
+The limiting error of the test in this case is again erfc
+�
+k2−k1
+4
+�
+V0
+2
+�
+as in Theorem 4.2, where V0 is
+defined by (4.19).
+4.3
+Test with entrywise transformation for spiked matrices of additive type
+The entrywise transform we applied with the PCA in Section 3.1 can also be adapted to be used together
+with the proposed test in Algorithm 2; see also [22] where the same idea was applied for the rank-1 spiked
+Wigner matrix. Recall the transformation defined in (3.1) and the transformed matrix �
+M in (3.2). We
+consider a test statistic
+�Lω := − log det
+�
+(1 + ωFg)I −
+�
+ωFg �
+M
+�
++ ωFg
+2 N
++ √ω
+�
+2
+�
+Fg,d
+w2
+−
+�
+Fg
+�
+Tr �
+M + λ
+� GH
+�
+w4 − 1 − Fg
+2
+�
+(Tr �
+M 2 − N),
+(4.21)
+where
+GH =
+1
+2Fg
+� ∞
+−∞
+g′(w)2g′′(w)
+g(w)2
+dw,
+�
+w4 =
+1
+(Fg)2
+� ∞
+−∞
+(g′(w))4
+(g(w))3 dw.
+We then have the following CLT result for �Lω that generalizes the results in [22].
+Theorem 4.5. Assume the conditions in Theorem 4.1, satisfying Assumption 3.1 with φ > 3/8. If λFg < 1,
+�Lω ⇒ N( �mk, �V0),
+(4.22)
+where the mean and the variance are given by
+�mk = −1
+2 log(1 − ωFg) +
+�(w2 − 1)GH
+�w4 − 1
+− Fg
+2
+�
+ω + �w4 − 3
+4
+(ωFg)2
++ k
+�
+− log(1 − ωFg) +
+�2Fg,d
+w2
+− Fg
+�
+ω +
+� (GH)2
+�w4 − 1 − (Fg)2
+2
+�
+ω2
+�
+,
+(4.23)
+�V0 = −2 log(1 − ωFg) +
+�4Fg,d
+w2
+− 2Fg
+�
+ω +
+�2(GH)2
+�w4 − 1 − (Fg)2
+�
+ω2.
+(4.24)
+16
+
+Algorithm 3 Hypothesis test for a spiked Wigner matrix with entrywise transformation
+Data: Mij, parameters w2, w4, λ, densities g, gd
+�
+M ← transformed matrix in (3.2),
+�Lω ← test statistic in (4.21),
+�m(k1+k2)/2 ← critical value in (4.25)
+with (4.23)
+if �Lω ≤ �m(k1+k2)/2 then
+Accept H1
+else
+Accept H2
+end if
+Proof. Theorem 4.5 directly follows from Theorem 5.3 in Section 5.
+Based on Theorem 4.5, we can adapt the test in Algorithm 2 to construct a test that utilizes the entrywise
+transformation. In this test, we compute �LΛ and compare it with the critical value
+�m(k1+k2)/2 := ( �mk1 + �mk2)/2.
+(4.25)
+See Algorithm 3 for the detail. The limiting error of the test is given as follows.
+Theorem 4.6. The error of the test in Algorithm 3 converges to
+erfc
+�
+�k2 − k1
+4
+�
+�V0
+2
+�
+� .
+Proof. Theorem 4.6 is a direct consequence of Theorem 5.6.
+We also propose an analogous test can for the additive model of the spiked rectangular matrices as follows.
+Recall the transformed matrix �Y ≡ �Y (0) in (3.3). Define the test statistic �Lω by
+�Lω = − log det
+��
+1 + d0
+ωFg
+�
+(1 + ωFg)I − �Y �Y T
+�
++ 2ω
+d0
+� GH
+�w4 − 1 − Fg
+2
+�
+(Tr �Y �Y T − M)
++ M
+�ωFg
+d0
+− log
+�ωFg
+d0
+�
+− 1 − d0
+d0
+log(1 + ωFg)
+�
+.
+(4.26)
+We then have the following CLT for the test statistic.
+Theorem 4.7. Assume the conditions in Theorem 4.4, satisfying Assumption 3.1 with φ > 3/8. If λ <
+√d0/Fg,
+�Lω ⇒ N( �mk, �V0),
+(4.27)
+where the mean and the variance are given by
+�m0 = −1
+2 log
+�
+1 − ω2(Fg)2
+d0
+�
++ ω2(Fg)2
+2d0
+( �w4 − 3)
+(4.28)
+�mk = �m0 + k
+�
+− log
+�
+1 − ω2(Fg)2
+d0
+�
++ 2ω2
+d0
+� (GH)2
+�w4 − 1 − (Fg)2
+2
+��
+(4.29)
+and
+�V0 = 4ω2
+d0
+� (GH)2
+�w4 − 1 − (Fg)2
+2
+�
+− 2 log
+�
+1 − ω2(Fg)2
+d0
+�
+.
+(4.30)
+17
+
+With Theorem 4.7, we can adjust Algorithm 2 for the weak detection of the signal in the additive model
+of spiked rectangular matrices, where we make the following change:
+• Data matrix is Yij (instead of Mij).
+• Transformed matrix is �Y (instead of �
+M), defined by (3.3) with α = 0.
+• Test statistic �Lω is defined by (4.26) (instead of (4.21)).
+• Critical value m(k1+k2)/2 is obtained by (4.25) with (4.29) (instead of (4.23)).
+In Appendix A, we consider several examples of spiked Wigner matrices and spiked rectangular matrices,
+where we compare the errors from numerical simulations and the theoretical errors of the proposed algorithms.
+We find that the numerical errors of the proposed tests closely match the corresponding theoretical errors
+and the error from Algorithm 3 is lower than that of Algorithm 2.
+4.4
+Rank estimation
+The test in Algorithm 2 requires prior knowledge about k1 and k2, the possible ranks of the planted spike.
+In this section, we adapt the idea of the proposed tests in Algorithm 2 to estimate the rank of the signal
+when there is no prior information on the rank k. Recall that the test statistic Lω defined in (4.9) does not
+depend on the rank of the matrix. As proved in Theorem 4.1, the test statistic Lω converges to a Gaussian
+random variable with mean mk and the variance V0, where mk is equi-distributed with respect to k and V0
+does not depend on k. It is then natural to set the best candidate for k, which we call κ, be the minimizer
+of the distance |Lω − mk|. This procedure is equivalent to find the nearest nonnegative integer of the value
+κ′ := 2(Lω − m0)
+V0
+(4.31)
+rounding half down.
+We describe the procedure in Algorithm 4; for example, its probability of error for spiked Wigner matrix
+converges to
+P(k = 0) · P
+�
+Z >
+√V0
+4
+�
++
+∞
+�
+i=1
+P(k = i) · P
+�
+|Z| >
+√V0
+4
+�
+=
+�
+1 − P(k = 0)
+2
+�
+· erfc
+�
+1
+4
+�
+V0
+2
+�
+,
+(4.32)
+where Z is a standard Gaussian random variable. Note that it depends only on P(k = 0).
+The error can be lowered if the range of k is known a priori. See Appendix A. It is also possible to
+improve Algorithm 4 by pre-transforming the data matrix entrywise as in Section 4.3. We omit the detail.
+5
+Central Limit Theorems
+In this section, we collect our results on general CLTs for the LSS of spiked random matrices. To precisely
+define the statements, we introduce the Chebyshev polynomials of the first kind.
+Definition 5.1 (Chebyshev polynomial). The n-th Chebyshev polynomial (of the first kind) Tn is a degree
+n polynomial defined by T0(x) = 1, T1(x) = x, and
+Tn+1(x) = 2xTn(x) − Tn−1(x).
+18
+
+Algorithm 4 Rank estimation
+Data: Mij (or Yij), parameters w2, w4, λ
+Lω ← test statistic in (4.9) or (4.16),
+m0 ← mean in (4.10) or (4.20),
+m1 ← mean in (4.13) or (4.18)
+with k = 1
+κ′ ← value in (4.31)
+if Lω ≤ (m0 + m1)/2 then
+Set κ = 0
+else
+Set κ = ⌈κ′ − 0.5⌉
+end if
+We first state a CLT for the LSS of spiked Wigner matrices. Recall that we denote by µ1 ≥ µ2 ≥ · · · ≥ µN
+the eigenvalues of a spiked Wigner matrix M.
+Theorem 5.2. Assume the conditions in Theorem 4.1. Suppose that a function f is analytic on an open
+interval containing [−2, 2]. Then,
+� N
+�
+i=1
+f(µi) − N
+� 2
+−2
+√
+4 − z2
+2π
+f(z) dz
+�
+⇒ N (mk(f), V0(f)) .
+The mean and the variance of the limiting Gaussian distribution are given by
+mk(f) = 1
+4 (f(2) + f(−2)) − 1
+2τ0(f) + (w2 − 2)τ2(f) + (w4 − 3)τ4(f) + k
+∞
+�
+ℓ=1
+√
+ωℓτℓ(f),
+V0(f) = (w2 − 2)τ1(f)2 + 2(w4 − 3)τ2(f)2 + 2
+∞
+�
+ℓ=1
+ℓτℓ(f)2 ,
+where we let
+τℓ(f) = 1
+π
+� 2
+−2
+Tℓ
+�x
+2
+�
+f(x)
+√
+4 − x2 dx.
+Furthermore, for mk, m0, and V0 defined in Theorem 4.1,
+�����
+mk(f) − m0(f)
+�
+V0(f)
+����� ≤
+����
+mk − m0
+√V0
+����
+The equality holds if and only if f(x) = C1φω(x) + C2 for some constants C1 and C2 where
+φω(x) := log
+�
+1
+1 − √ωx + ω
+�
++ √ω
+� 2
+w2
+− 1
+�
+x + ω
+�
+1
+w4 − 1 − 1
+2
+�
+x2.
+We will give a proof of Theorem 5.2 in Appendix C. With the entrywise transformation in Section 4.3,
+we have the following changes in Theorem 5.2. Recall that �µ1 ≥ �µ2 ≥ · · · ≥ �µN are the eigenvalues of the
+transformed matrix �
+M.
+Theorem 5.3. Assume the conditions in Theorem 5.2, satisfying Assumption 3.1 with φ > 3/8. If λFg < 1,
+� N
+�
+i=1
+f(�µi) − N
+� 2
+−2
+√
+4 − z2
+2π
+f(z) dz
+�
+⇒ N( �mk(f), �V0(f)) .
+19
+
+The mean and the variance of the limiting Gaussian distribution are given by
+�mk(f) = 1
+4 (f(2) + f(−2)) − 1
+2τ0(f) + k
+�
+ωFg,dτ1(f) + (w2 − 2 + kωGH)τ2(f)
++ (�
+w4 − 3)τ4(f) + k
+∞
+�
+ℓ=3
+�
+(ωFg)ℓτℓ(f),
+(5.1)
+�V0(f) = (w2 − 2)τ1(f)2 + 2(�
+w4 − 3)τ2(f)2 + 2
+∞
+�
+ℓ=1
+ℓτℓ(f)2.
+Furthermore, for �mk, �m0, and �V0 defined in Theorem 4.1,
+������
+�mk2(f) − �mk1(f)
+�
+�V0(f)
+������
+≤
+������
+�mk2 − �mk1
+�
+�V0
+������
+The equality holds if and only if f(x) = C1 �φω(x) + C2 for some constants C1 and C2 with the function
+�φω(x) := log
+�
+1
+1 −
+�
+ωFgx + ωFg
+�
++
+�
+2
+�
+Fg,d
+w2
+−
+�
+Fg
+�
+x + ω
+� GH
+�w4 − 1 − Fg
+2
+�
+x2.
+We will also prove Theorem 5.3 in Appendix C.
+Remark 5.4. For a general case where the spike Λ = diag(ω1, · · · , ωk) with possibly distinct ωi’s, we can
+prove the CLT and the transformed CLT, analogous to Theorems 5.2 and 5.3, respectively, where the means
+of the limiting Gaussians are given by
+mM(f) = 1
+4 (f(2) + f(−2)) − 1
+2τ0(f) + (w2 − 2)τ2(f) + (w4 − 3)τ4(f)
++
+k
+�
+s=1
+∞
+�
+ℓ=1
+�
+ωℓsτℓ(f),
+�mM(f) = 1
+4 (f(2) + f(−2)) − 1
+2τ0(f) + (w2 − 2)τ2(f) + (�
+w4 �� 3)τ4(f)
++
+k
+�
+s=1
+�
+ωsFg,dτ1(f) + ωsGHτ2(f) +
+k
+�
+s=1
+∞
+�
+ℓ=3
+�
+(ωsFg)ℓτℓ(f),
+and the variances are equal to V0(f) in Theorem 5.2 and �V0(f) in Theorem 5.3, respectively. Adapting the
+proposed tests in Algorithms 2 and 3, it is possible to construct hypothesis tests for the weak detection in this
+case.
+The next result is the CLT for the LSS of spiked rectangular matrices Y , where we denote by µ1 ≥ µ2 ≥
+· · · ≥ µM the eigenvalues of Y Y T .
+Theorem 5.5. Assume the conditions in Theorem 4.4. Suppose that a function f is analytic on an open
+set containing an interval [d−, d+]. Then,
+� M
+�
+i=1
+f(µi) − M
+� d+
+d−
+�
+(x − d−)(d+ − x)
+2πd0x
+f(x) dx
+�
+⇒ N(mk(f), V0(f)).
+(5.2)
+20
+
+The mean and the variance of the limiting Gaussian distribution are given by
+mk(f) =
+�f(2) + �f(−2)
+4
+− τ0( �f)
+2
++ (w4 − 3)τ2( �f) + k
+∞
+�
+ℓ=1
+� ω
+√d0
+�ℓ
+τℓ( �f)
+and
+V0(f) = 2
+∞
+�
+ℓ=1
+ℓτℓ( �f)2 + (w4 − 3)τ1( �f)2,
+where we let �f(x) = f(√d0x + 1 + d0).
+Furthermore, for mk, m0, and V0 defined in Theorem 4.4,
+�����
+mk2(f) − mk1(f)
+�
+V0(f)
+����� ≤
+����
+mk2 − mk1
+√V0
+����
+The equality holds if and only if f(x) = C1φω(x) + C2 for some constants C1 and C2 with the function
+φω(x) = ω
+d0
+�
+2
+w4 − 1 − 1
+�
+x − log
+��
+1 + d0
+ω
+�
+(1 + ω) − x
+�
+.
+Lastly, we state the pre-transformed CLT for the LSS of the additive model of spiked rectangular matrices.
+We let �Y be the transformed matrix and �µ1 ≥ �µ2 ≥ · · · ≥ �µN the eigenvalues of �Y �Y T .
+Theorem 5.6. Assume the conditions in Theorem 5.5, satisfying Assumption 3.1 with φ > 3/8. If λ <
+√d0/Fg,
+� M
+�
+i=1
+f(�µi) − M
+� d+
+d−
+f(x)ρMP,d0(dx)
+�
+⇒ N( �mk(f), �V0(f)).
+(5.3)
+The mean and the variance of the limiting Gaussian distribution are given by
+�mk(f) =
+�f(2) + �f(−2)
+4
+− 1
+2τ0( �f) + kω
+√d0
+(GH − Fg)τ1( �f) + (�
+w4 − 3)τ2( �f)
++ k
+∞
+�
+ℓ=1
+�ωFg
+√d0
+�ℓ
+τℓ( �f)
+(5.4)
+and
+�V0(f) = 2
+∞
+�
+ℓ=1
+ℓτℓ( �f)2 + (�
+w4 − 3)τ1( �f)2.
+(5.5)
+where �f(x) = f(√d0x + 1 + d0).
+Furthermore, for �mk, �m0, and �V0 defined in Theorem 4.7, The equality holds if and only if f(x) =
+C1 �φω(x) + C2 for some constants C1 and C2 with the function
+�φω(x) = 2λ
+d0
+� GH
+�w4 − 1 − Fg
+2
+�
+x − log
+�� d0
+ωFg
++ 1
+�
+(ωFg + 1) − x
+�
+.
+Remark 5.7. As in Remark 5.4, for a general case with Λ = diag(ω1, · · · , ωk), the CLT and the transformed
+21
+
+CLT hold with the adjusted means
+mY (f) =
+�f(2) + �f(−2)
+4
++ τ0( �f)
+2
++ (w4 − 3)τ2( �f) +
+k
+�
+s=1
+∞
+�
+ℓ=1
+� ωs
+√d0
+�ℓ
+τℓ( �f),
+�mY (f) =
+�f(2) + �f(−2)
+4
+− 1
+2τ0( �f) + (�
+w4 − 3)τ2( �f) +
+k
+�
+s=1
+ωs
+√d0
+(GH − Fg)τ1( �f)
++
+k
+�
+s=1
+∞
+�
+ℓ=1
+�ωsFg
+√d0
+�ℓ
+τℓ( �f),
+where the variances are given V0(f), �V0(f), respectively. Further, the corresponding optimal functions and
+test statistic can be calculated by following the same procedure in [34].
+6
+Conclusion and Future Works
+In this paper, we considered the detection problems of the spiked random model with general ranks. First,
+we prove the sub-optimality of the PCA for the non-Gaussian noise. Further, we proposed a hypothesis test
+based on the central limit theorem for the linear spectral statistics of the data matrix and introduced a test
+for rank estimation that do not require any prior information on the rank of the signal. It was shown that
+the error of the proposed hypothesis test matches the error of the likelihood ratio test in case the noise is
+Gaussian and the signal-to-noise ratio is small. With the knowledge on the density of the noise, the test was
+further improved by applying an entrywise transformation.
+We believe that the hypothesis test with the entrywise transformed matrix proposed in this paper can
+be extended to the multiplicative model of spiked rectangular matrix. This will be discussed in our future
+works.
+Acknowledgments
+The work of J. H. Jung and J. O. Lee was partially supported by National Research Foundation of Korea
+under grant number NRF-2019R1A5A1028324.
+The work of H. W. Chung was partially supported by
+National Research Foundation of Korea under grant number 2017R1E1A1A01076340 and by the Ministry
+of Science and ICT, Korea, under an ITRC Program, IITP-2019-2018-0-01402.
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+A
+Examples and Simulations
+In Appendix A, we consider specific examples of spiked random matrices under various settings. We first
+demonstrate with an example the change of the threshold by the improved PCA in Section 3. We then
+provide the details of the proposed tests in Algorithms 2 and 3 with different examples, and the test for
+rank estimation in Algorithm 4 for these and compute the theoretical errors. We also perform the numerical
+simulation for the proposed tests and compare the numerical errors with the theoretical errors.
+A.1
+Spiked Wigner matrix
+A.1.1
+Improved PCA with Entrywise Transformation
+Our first example is a spiked Wigner matrix with non-Gaussian noise to which we apply the entrywise
+transformation for the improved PCA. We let the density function of the noise be a bimodal distribution
+with unit variance, defined as
+g(x) = gd(x) =
+1
+√
+2π
+�
+e−2(x−
+√
+3/2)2 + e−2(x+
+√
+3/2)2�
+,
+(A.1)
+25
+
+which is the density function of a random variable
+1
+2N +
+√
+3
+2 R,
+where N is a standard Gaussian random variable and R is a Rademacher random variable, independent to
+each other.
+We sample Zij = Zji independently from the density g and let Wij = Zij/
+√
+N.
+We let u(ℓ) =
+(u1(ℓ), u2(ℓ), . . . , uN(ℓ))T , where
+√
+Nui(ℓ)’s are i.i.d. Rademacher random variables for i = 1, 2, . . . , N and
+ℓ = 1, 2, 3. The data matrix M = UΛ1/2U T , where U = [u(1), u(2), u(3)] and Λ = diag(λ, λ, λ, 0, 0, . . . , 0).
+The size of the data matrix is set to be N = 4000. The BBP-transition predicts that the largest eigenvalue
+of M pops up from the bulk of the spectrum if λ > 1.
+With the entrywise transformation defined in (3.2), we obtain a transformed matrix
+�
+Mij =
+1
+�
+FgN h(
+√
+NMij)
+(A.2)
+where
+h(x) = −g′(x)
+g(x) =
+2
+�√
+3 − e4
+√
+3x(
+√
+3−2x) + 2x
+�
+1 + e4
+√
+3x
+(A.3)
+and Fg =
+� ∞
+−∞
+(g′(x))2
+g(x) dx ≈ 2.50810. From Theorem 3.3, it is expected that the largest eigenvalue of �
+M
+separates from other eigenvalues if λ >
+1
+Fg ≈ 0.3987.
+In the numerical experiment, we set
+λℓ =
+ℓ +
+1
+Fg
+ℓ + 1
+(A.4)
+for ℓ = 1, 2, 3, and we compare the spectrum of the matrices M and �
+M. In Figure 1, we find three isolated
+eigenvalues in the spectrum of �
+M (right), which are absent in that of M (left).
+-2
+-1.5
+-1
+-0.5
+0
+0.5
+1
+1.5
+2
+0
+5
+10
+15
+20
+25
+30
+-2
+-1.5
+-1
+-0.5
+0
+0.5
+1
+1.5
+2
+0
+5
+10
+15
+20
+25
+30
+Figure 1: The spectrum of the data matrix (N = 4000) with bimodal noise, before (left) and after (right)
+the entrywise transformation. Three eigenvalues pop up from the bulk of the spectrum after the entrywise
+transformation.
+26
+
+A.1.2
+Spiked Gaussian Wigner matrix
+We consider the weak detection problem with the simplest case of the spiked Gaussian Wigner matrix where
+w2 = 2 (i.e., W is a GOE matrix) and the signal u(m) = (u1(m), u2(m), . . . , uN(m)) where
+√
+Nui(m)’s are
+i.i.d. Rademacher random variable. Note that the parameters w2 = 2 and w4 = 3.
+In the numerical simulation done in Matlab, we generated 10,000 independent samples of the 256 × 256
+data matrix M, where we fix k1 = 1 (under H1) and vary k2 from 2 to 5 (under Hk2), with the SNR λ
+varying from 0 to 0.7. To apply Algorithm 2, we compute
+Lλ = − log det
+�
+(1 + λ)I −
+√
+λM
+�
++ λN
+2 .
+(A.5)
+We accept H1 if
+Lλ ≤ mk1 + mk2
+2
+= −k2 + 2
+2
+log(1 − λ)
+and reject H1 otherwise. The (theoretical) limiting error of the test is
+erfc
+�k2 − 1
+4
+�
+− log(1 − λ)
+�
+.
+(A.6)
+In Figure 2, we compare the error from the numerical simulation and the theoretical error of the proposed
+algorithm, which show that the numerical errors of the test closely match the theoretical errors.
+Figure 2: The errors from the simulation with Algorithm 2 (solid) versus the limiting errors (A.6) (dashed)
+for the setting in Section A.1.2 with k2 = 2, 3, 4, 5.
+A.1.3
+Spiked Wigner matrix
+We next consider a spiked Wigner matrix with non-Gaussian noise, where the density function of the noise
+matrix is given by
+g(x) = gd(x) =
+1
+2 cosh(πx/2) =
+1
+eπx/2 + e−πx/2 .
+(A.7)
+27
+
+0.9
+0.8
++ Type II error
+0.7
+0.6
+0.5
+k.=2
+(,=2 (limiting)
+≥ 0.4
+k,=3
+k,=3 (limiting)
+0.3
+=4 (limiting)
+0.2
+k,=5 (limiting)
+0.1
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+rWe sample Zij = Zji from the density g and let Wij = Zij/
+√
+N.
+We again let the signal u(m) =
+(u1(m), u2(m), . . . , uN(m)) where
+√
+Nui(m)’s are i.i.d. Rademacher random variable. Note that the pa-
+rameters w2 = 1 and w4 = 5. We again perform the numerical simulation 10,000 samples of the 256 × 256
+data matrix M with the SNR λ varying from 0 to 0.6, where we fix k1 = 1 (under H1) and k2 = 3 (under
+H2).
+In Algorithm 2, we compute
+Lλ = − log det
+�
+(1 + λ)I −
+√
+λM
+�
++ λN
+2
++
+√
+λ Tr M − λ
+4 (Tr M 2 − N).
+(A.8)
+We accept H1 if
+Lλ ≤ mk1 + mk2
+2
+= −k2 + 2
+2
+log(1 − λ) + k2λ
+2
+− (k2 − 3)λ2
+8
+and accept H2 otherwise. The (theoretical) limiting error of the test is
+erfc
+�
+k2 − 1
+4
+�
+− log(1 − λ) + λ − λ2
+4
+�
+.
+(A.9)
+We can further improve the test by introducing the entrywise transformation given by
+h(x) = −g′(x)
+g(x) = π
+2 tanh πx
+2 .
+The Fisher information Fg = π2
+8 , which is larger than 1. We thus construct a transformed matrix �
+M by
+�
+Mij = 2
+√
+2
+π
+√
+N
+h(
+√
+NMij) =
+�
+2
+N tanh
+�
+π
+√
+N
+2
+Mij
+�
+.
+If λ >
+1
+Fg =
+8
+π2 ≈ 0.8106, we can apply PCA for strong detection of the signal. If λ <
+8
+π2 , applying Algorithm
+3, we compute
+�Lλ = − log det
+��
+1 + π2λ
+8
+�
+I −
+�
+π2λ
+8
+�
+M
+�
++ π2λN
+16
++ π
+√
+λ
+2
+√
+2 Tr �
+M + π2λ
+16 (Tr �
+M 2 − N).
+(Here, Fg = Fg,d = π2
+8 , GH = π2
+16 , and �w4 = 3
+2.) We accept H1 if
+�Lλ ≤ −k2 + 2
+2
+log
+�
+1 − π2λ
+8
+�
++ k2π2λ
+16
+− 3π4λ2
+512
+and accept H2 otherwise. The limiting error with entrywise transformation is
+erfc
+�
+k2 − 1
+4
+�
+− log
+�
+1 − π2λ
+8
+�
++ π2λ
+8
+�
+.
+(A.10)
+Since erfc(·) is a decreasing function and π2
+8 > 1, it is immediate to see that the limiting error in (A.10) is
+strictly smaller than the limiting error in (A.9).
+In Figure 3, we plot the result of the simulation with k2 = 3, which shows that the numerical error from
+Algorithm 3 is smaller than that of Algorithm 2; both errors closely match theoretical errors in (A.10) and
+(A.9).
+28
+
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1
+Type I + Type II error
+Alg.2 k2=3
+Alg.2 k2=3(limiting)
+Alg.3 k2=3
+Alg.3 k2=3(limiting)
+Figure 3: The errors from the simulation with Algorithm 2 (blue) and with Algorithm 3 (yellow), respectively,
+versus the limiting errors (A.9) of Algorithm 2 (red) and (A.10) of Algorithm 3 (purple), respectively, for
+the setting in Section A.1.3.
+A.1.4
+Rank Estimation
+We again consider the example in Section A.1.2 and apply Algorithm 4 to estimate the rank of the signal.
+We again perform the numerical simulation 20,000 samples of the 256 × 256 data matrix M with the SNR
+λ varying 0.025 to 0.6 and choose the rank of the signal k uniformly from 0 to 4. Since we know that the
+range of the rank k is [0, 4], the (theoretical) limiting error in (4.32) changes to
+P(k = 0) · P
+�
+Z >
+√V0
+4
+�
++
+3
+�
+i=1
+P(k = i) · P
+�
+|Z| >
+√V0
+4
+�
++ P(k = 4) · P
+�
+Z >
+√V0
+4
+�
+=
+�
+1 − P(k = 0) + P(k = 4)
+2
+�
+× erfc
+�
+1
+4
+�
+− log(1 − λ) +
+� 2
+w2
+− 1
+�
+λ +
+�
+1
+w4 − 1 − 1
+2
+�
+λ2
+�
+.
+We compute the same test statistic
+Lλ = − log det
+�
+(1 + λ)I −
+√
+λM
+�
++ λN
+2
+(A.11)
+and find the nearest nonnegative integer of the value
+−
+Lλ
+log(1 − λ) − 1
+2,
+(A.12)
+rounding half down. Since P(k = 0) = P(k = 4) = 0.2, the limiting error of the estimation is
+�
+1 − P(k = 0) + P(k = 4)
+2
+�
+· erfc
+�1
+4
+�
+− log(1 − λ)
+�
+= 0.8 · erfc
+�1
+4
+�
+− log(1 − λ)
+�
+.
+(A.13)
+29
+
+The result of the simulation can be found in Figure 4, where we compare the error from the estimation
+(Algorithm 4) and the theoretical error in (A.13). We can see that the error from the numerical simulation
+matches closely the theoretical error.
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.6
+0.62
+0.64
+0.66
+0.68
+0.7
+0.72
+0.74
+0.76
+Probability of error
+Alg.4
+limiting error
+Figure 4: The errors from the simulation with Algorithm 4 (solid) versus the limiting error (A.13) (dashed)
+for the setting in Section A.1.4.
+A.2
+Spiked rectangular matrices
+In this section, we check the performance of the improved PCA and the pre-transformed LSS-based tests for
+spiked rectangular matrices.
+A.2.1
+Improved PCA with Entrywise Transformation
+Additive model
+We consider the data with the non-Gaussian noise whose density function is given by the bimodal dis-
+tribution in (A.1).
+We sample Zij independently from the density g and let Xij = Zij/
+√
+N.
+We let
+u(ℓ) = (u1(ℓ), u2(ℓ), . . . , uM(ℓ))T and v(ℓ) = (v1(ℓ), v2(ℓ), . . . , vN(ℓ))T , where
+√
+Mui(ℓ)’s and
+√
+Nvj(ℓ)’s are
+i.i.d. Rademacher random variables for i = 1, 2, . . . , M, j = 1, 2, . . . , N and ℓ = 1, 2, 3. When we apply the
+entrywise transformation, defined in (3.3), with α = 0 to the rank-3 spiked mean data matrix, we get
+�Yij =
+1
+�
+FgN h(
+√
+NYij)
+(A.14)
+where
+h(x) = −g′(x)
+g(x) =
+2
+�√
+3 − e4
+√
+3x(
+√
+3−2x) + 2x
+�
+1 + e4
+√
+3x
+(A.15)
+and Fg =
+� ∞
+−∞
+(g′(x))2
+g(x) dx ≈ 2.50810. The size of the data matrix is set to be M = 2000, N = 4000, and the
+ratio d0 = M/N = 0.5.
+30
+
+Theoretically, the threshold for the BBP-transition of the largest eigenvalue is √d0 ≈ 0.7071 with the
+vanilla PCA, whereas the threshold is lowered to
+√d0
+Fg
+≈ 0.2819 with the improved PCA as predicted by
+Theorem 3.4.
+For ℓ = 1, 2, 3, we set the SNRs
+λℓ =
+ℓ√d0 +
+√d0
+Fg
+ℓ + 1
+(A.16)
+to observe the transitions of the largest eigenvalue after the transformation. In Figure 5, we compare the
+spectrum of the sample covariance matrices, Y Y T (left) and �Y �Y T (right). As in the spiked Wigner case in
+Section A.1.1, we again find three outlier eigenvalues only in the spectrum of �Y �Y T (right), which are absent
+in that of Y Y T (left).
+0
+1
+2
+3
+0
+5
+10
+15
+20
+25
+30
+35
+40
+0
+1
+2
+3
+0
+5
+10
+15
+20
+25
+30
+35
+40
+Figure 5: The spectrum of the sample covariance matrix (M = 2000, N = 4000) with bimodal noise, before
+(left) and after (right) the entrywise transformation. Three eigenvalues pop up from the bulk of the spectrum
+after the entrywise transformation.
+Multiplicative model
+In the spiked covariance model, to clearly observe the outlier in our simulation setting, a distribution with
+a larger Fisher information value should be used. Thus, we let the density function ga of the noise be the
+generalized version of the bimodal distribution with unit variance in (A.1), defined as
+ga(x) =
+1
+2
+�
+2(1 − a2)π
+�
+e
+− (x−a)2
+2(1−a2) + e
+− (x+a)2
+2(1−a2)
+�
+,
+which ga is the density function of a random variable
+�
+1 − a2N + aR.
+31
+
+We sample Zij independently from the density ga and let Xij = Zij/
+√
+N. We let u(ℓ) = (u1(ℓ), u2(ℓ), . . . , uM(ℓ))T
+and v(ℓ) = (v1(ℓ), v2(ℓ), . . . , vN(ℓ))T , where
+√
+Mui(ℓ)’s and
+√
+Nvj(ℓ)’s are i.i.d. Rademacher random vari-
+ables for i = 1, 2, . . . , M, j = 1, 2, . . . , N and ℓ = 1, 2, 3. When we apply the entrywise transformation,
+defined in (3.3) to the rank-3 spiked covariance data matrix, we get
+�Yij =
+1
+�
+(α2 + 2α + Fg)N
+ha,α(
+√
+NYij)
+(A.17)
+where
+ha,α(x) = −g′
+a(x)
+ga(x) + αx =
+�
+(x − a)e
+2ax
+1−a2 + (x + a)
+�
+(1 − a2)(1 + e
+2ax
+1−a2 )
++ αx
+(A.18)
+and Fg =
+� ∞
+−∞
+(g′
+a(x))2
+ga(x) dx ≈ 5.15583, when a =
+√
+21/5. The size of the data matrix is set to be M = 4000,
+N = 8000. We also use α =
+�
+Fg, and the ratio d0 = M/N = 0.5. The threshold for the BBP-transition
+of the largest eigenvalue is √d0 ≈ 0.7071 for the vanilla PCA, whereas the threshold changes to λg,ℓ =
+(1+√
+Fg)
+2
+· (2γℓ +
+�
+Fgγ2
+ℓ ) = √d0 for the transformed PCA. (See Theorem 3.5.) For ℓ = 1, 2, 3, we set the
+SNRs
+λℓ =
+ℓ√d0 +
+2√d0
+1+√
+Fg
+ℓ + 1
+(A.19)
+to observe the transitions of the largest eigenvalue after the transformation.
+We obtain a result analogous to the additive model. See Figure 6.
+0
+1
+2
+3
+0
+10
+20
+30
+40
+50
+60
+70
+80
+0
+1
+2
+3
+0
+10
+20
+30
+40
+50
+60
+70
+80
+Figure 6: The spectrum of the sample covariance matrix (M = 4000, N = 8000) with bimodal noise, before
+(left) and after (right) the entrywise transformation. Three eigenvalues pop up from the bulk of the spectrum
+after the entrywise transformation.
+32
+
+A.2.2
+Hypothesis Testing with pre-transformed LSS estimator
+We now consider an (additive) spiked rectangular matrix with the non-Gaussian noise whose density function
+is given by (A.7). We let the signal u = (u1, u2, . . . , uM)T and v = (v1, v2, . . . , vN)T , where
+√
+Mui’s and
+√
+Nvj’s are i.i.d. Rademacher random variables for i = 1, 2, . . . , M and j = 1, 2, . . . , N. Let the data matrix
+Y =
+√
+λuvT + X.
+Recall that w4 = 5, Fg = π2
+8 , GH = π2
+16 , and �w4 = 3
+2. The LSS estimators are given by
+Lω = − log det
+��
+1 + d0
+ω
+�
+(1 + ω)I − Y Y T
+�
+− ω
+2d0
+(Tr Y Y T − M)
++ M
+� ω
+d0
+− log
+� ω
+d0
+�
+− 1 − d0
+d0
+log(1 + ω)
+�
+,
+(A.20)
+and
+�Lω = − log det
+��
+1 + 8d0
+ωπ2
+�
+(1 + ω π2
+8 )I − �Y �Y T
+�
++ π2ω
+8d0
+(Tr �Y �Y T − M)
++ M
+�ωπ2
+8d0
+− log
+�ωπ2
+8d0
+�
+− 1 − d0
+d0
+log
+�
+1 + ω π2
+8
+��
+.
+(A.21)
+With critical values mω = − log
+�
+1 − ω2
+d0
+�
++ 3ω2
+4d0 and �mω = − log
+�
+1 − ω2π4
+64d0
+�
+− 3π4ω2
+256d0, the errors are
+erfc
+�
+1
+4
+�
+− log
+�
+1 − ω2
+d0
+�
+− ω2
+2d0
+�
+and
+erfc
+�
+1
+4
+�
+− log
+�
+1 − π4ω2
+64d0
+��
+.
+In Figure 7, we plot empirical average (after 1,000 Monte Carlo simulations) of the error of the proposed
+test and the theoretical (limiting) error, varying the SNR ω from 0 to 0.5, with M = 256 and N = 512. It
+can be checked that the error of the proposed test closely matches the theoretical error.
+0
+0.05
+0.1
+0.15
+0.2
+0.25
+0.3
+0.35
+0.4
+0.45
+0.5
+0.65
+0.7
+0.75
+0.8
+0.85
+0.9
+0.95
+1
+simulation
+limiting error
+simulation-transformed
+limiting error-tansformed
+Figure 7: The error from the simulation (solid) and the theoretical limiting error (dashed), respectively.
+33
+
+B
+Proof of Theorems for improved PCA
+In this section, we rigorously prove Theorems 3.4 and 3.5 in Section 3, which are about the detection threshold
+of the improved PCA.
+B.1
+Preliminaries
+We first introduce the following notions, which provide a simple way of making precise statements regarding
+the bound up to small powers of N that holds with probability higher than 1 − N −D for all D > 0.
+Definition B.1 (Overwhelming probability). We say that an event (or family of events) Ω holds with
+overwhelming probability if for all (large) D > 0 we have P(Ω) ≤ N −D for any sufficiently large N.
+Definition B.2 (Stochastic domination). Let
+ξ =
+�
+ξ(N)(u) : N ∈ N, u ∈ U (N)�
+,
+ζ =
+�
+ζ(N)(u) : N ∈ N, u ∈ U (N)�
+be two families of random variables, where U (N) is a possibly N-dependent parameter set. We say that ξ is
+stochastically dominated by ζ, uniformly in u, if for all (small) ϵ > 0 and (large) D > 0
+sup
+u∈U (N) P
+�
+|ξ(N)(u)| > N ϵζ(N)(u)
+�
+≤ N −D
+for any sufficiently large N ≥ N0(ε, D). Throughout this appendix, the stochastic domination will always be
+uniform in all parameters, including matrix indices and the spectral parameter z.
+We write ξ ≺ ζ or ξ = O≺(ζ), if ξ is stochastically dominated by ζ, uniformly in u.
+For a Wigner matrix W, we will use the following result for the resolvents, which is called an isotropic
+local semicircle law.
+Lemma B.3 (Isotropic local semicircle law). Suppose that z ∈ R outside an open interval containing [−2, 2].
+Let ssc(z) be the Stieltjes transform of the Marchenko–Pastur law, which is also given by
+ssc(z) = −z +
+√
+z2 − 4
+2
+.
+(B.1)
+Then,
+⟨u(ℓ1), (W − zI)−1u(ℓ2)⟩ = ssc(z)⟨u(ℓ1), u(ℓ2)⟩ + O≺(N − 1
+2 )
+See Theorem 2.3 of [36] (also Lemma 7.7 of [22]) for the proof of Lemma B.3.
+Further, for a rectangular matrix X, we will use the following analogous result for the resolvents, which
+is called an isotropic Marchenko–Pastur law.
+Lemma B.4 (Isotropic local Marchenko–Pastur law). Suppose that z ∈ R outside an open interval containing
+[d−, d+]. Let s(z) be the Stieltjes transform of the Marchenko–Pastur law, which is also given by
+s(z) = (1 − d0 − z) +
+�
+(1 − d0 − z)2 − 4d0z
+2d0z
+.
+(B.2)
+Then,
+⟨v(ℓ1), (XT X − zI)−1v(ℓ2)⟩ = −
+�
+1
+zs(z) + 1
+�
+⟨v(ℓ1), v(ℓ2)⟩ + O≺(N − 1
+2 )
+34
+
+and
+⟨XT u(ℓ1), (XT X − zI)−1XT u(ℓ2)⟩ = (zs(z) + 1)⟨u(ℓ1), u(ℓ2)⟩ + O≺(N − 1
+2 ).
+See Theorem 2.5 of [17] (also Lemma 3.7 of [18]) for the proof of Lemma B.4.
+The following concentration inequality will be frequently used in the proof, which is sometimes called the
+large deviation estimate in random matrix theory.
+Lemma B.5 (Large deviation estimate). Let
+�
+ξ(N)
+i
+�
+and
+�
+ζ(N)
+i
+�
+be independent families of random variables
+and
+�
+a(N)
+ij
+�
+and
+�
+b(N)
+i
+�
+be deterministic; here N ∈ N and i, j = 1, . . . , N.
+Suppose that complex-valued
+random variables ξ(N)
+i
+and ζ(N)
+i
+are independent and satisfy for all p ≥ 2 that
+Eξ = 0 ,
+E|ξ|p ≤
+Cp
+NBp−2
+(B.3)
+for some B ≤ N 1/2 and some (N-independent) constant Cp. Then we have the bounds
+�
+i
+biξi ≺
+� 1
+N
+�
+i
+|bi|2
+�1/2
++ maxi |bi|
+B
+,
+(B.4)
+�
+i,j
+aijξiζj ≺
+� 1
+N 2
+�
+i̸=j
+|aij|2
+�1/2
++ maxi̸=j |aij|
+B
++ maxi |aii|
+B2
+,
+(B.5)
+�
+i̸=j
+aijξiξj ≺
+� 1
+N 2
+�
+i̸=j
+|aij|2
+�1/2
++ maxi̸=j |aij|
+B
+.
+(B.6)
+If the coefficients a(N)
+ij
+and b(N)
+i
+depend on an additional parameter u, then all of these estimates are uniform
+in u, i.e. N0 = N0(ε, D) in the definition of ≺ depends not on u but only on the constant C from (B.3).
+If B = N 1/2, the bounds can further be simplified to
+�
+i
+biξi ≺
+� 1
+N
+�
+i
+|bi|2
+�1/2
+,
+�
+i,j
+aijξiζj ≺
+� 1
+N 2
+�
+i,j
+|aij|2
+�1/2
+,
+�
+i̸=j
+aijξiξj ≺
+� 1
+N 2
+�
+i̸=j
+|aij|2
+�1/2
+.
+(B.7)
+Proof. These estimates are an immediate consequence of Lemma 3.8 in [30].
+Finally, we recall that, for our prior, |⟨u(ℓ1), u(ℓ2)⟩ − δℓ1ℓ2|, |⟨v(ℓ1), v(ℓ2)⟩ − δℓ1ℓ2| ≺ N −φ.
+B.2
+Proof of Theorem 3.3
+We first prove the behavior of the k largest eigenvalues described in Section 2.2, which we will call the BBP
+result, in our setting, following the strategy of [14, 15].
+M − zI = W + UΛ1/2U T − zI
+= (W − zI)(I + (W − zI)−1(UΛ1/2U T )).
+(B.8)
+Thus, if z is an eigenvalue of M but not of W, then it satisfies
+det(I + (W − zI)−1UΛ1/2U T ) = 0,
+which also implies that −1 is an eigenvalue of
+T ≡ T(z) := (W − zI)−1UΛ1/2U T .
+35
+
+We then see that
+T(W − zI)−1u(ℓ) = (W − zI)−1UΛ1/2U T (W − zI)−1u(ℓ)
+=
+�
+λℓ⟨u(ℓ), (W − zI)−1u(ℓ)⟩(W − zI)−1u(ℓ) + O≺(N −φ).
+i.e., (W − zI)−1u(ℓ) must be a eigenvector for T with the corresponding eigenvalue −1. Thus, by Lemma
+B.3,
+�
+λℓssc(z) = −1 + O≺(N −φ).
+It is elementary to check that the solution of the above equation is z = √λℓ +
+1
+√λℓ + O≺(N −φ) if and only
+if λℓ > 1.
+We now turn to the proof of Theorem 3.3. For the spike ∥U∥∞ ≺ N −φ, suppose that a function q and
+its all derivatives are polynomially bounded in the sense of Assumption 3.1. Following the proof of Theorem
+4.8 in [50], we have the following local linear estimation of q(
+√
+NMij) by
+q(
+√
+NMij) = q(
+√
+NWij) +
+√
+λNuiuT
+j E[q′(
+√
+NWij)] + Rij,
+where the error Rij is negligible. Set
+Mq := E[q′(
+√
+NWij)],
+Vq := E[q(
+√
+NWij)2],
+�λ := λM 2
+q /Vq,
+and
+Qij :=
+1
+�
+NVq
+q(
+√
+NWij).
+Then the spectrum of the transformed matrix is determined by the matrix Q + U ˆΛ1/2U T . Since Q is also
+Wigner matrix with NE[Q2
+ij] = 1, by repeating the same process above, we get the result.
+B.3
+Proof of Theorem 3.4
+We first prove the behavior of the k largest eigenvalues described in Section 2.2, which we will again call the
+BBP result, in our setting, following the strategy of [14, 15]. Note that the k largest eigenvalue of Y Y T is
+equal to the k largest eigenvalues of Y T Y . Consider the identity
+Y T Y − zI = (X + UΛ1/2V T )T (X + UΛ1/2V T ) − zI = (XT X − zI)T(z)
+(B.9)
+where
+T ≡ T(z) := (XT X − zI)−1(XT UΛ1/2V T + V Λ1/2U T X + V Λ1/2U T UΛ1/2V T ).
+Thus, if z is an eigenvalue of Y Y T but not of XXT , then it satisfies
+det(T(z)) = 0,
+which also implies that −1 is an eigenvalue of T(z).
+36
+
+Note that since ∥X∥, ∥(XT X − zI)−1∥ ≺ 1, from Lemma B.5,
+⟨b, (XT X − zI)−1XT a⟩ =
+�
+i,j
+�
+(XT X − zI)−1XT �
+ij biaj
+≺
+�
+� 1
+N 2
+�
+i̸=j
+���
+�
+(XT X − zI)−1XT �
+ij
+���
+2
+�
+�
+1/2
++ N −φ max
+i,j
+���
+�
+(XT X − zI)−1XT �
+ij
+���
+≺
+� 1
+N ∥(XT X − zI)−1XT ∥2
+�1/2
++ N −φ∥(XT X − zI)−1XT ∥ ≺ N −φ.
+Then the matrix T satisfies
+T · (XT X − zI)−1XT u(ℓ)
+= (XT X − zI)−1XT UΛ1/2(V T (XT X − zI)−1XT u(ℓ))
++ (XT X − zI)−1V Λ1/2(U T X(XT X − zI)−1XT u(ℓ))
++ (XT X − zI)−1V Λ1/2(U T U)Λ1/2(V T (XT X − zI)−1XT u(ℓ))
+=
+�
+λℓ⟨u(ℓ), X(XT X − zI)−1XT u(ℓ)⟩(XT X − zI)−1v(ℓ) + θ1(ℓ)
+and
+T · (XT X − zI)−1v(ℓ)
+= (XT X − zI)−1XT UΛ1/2(V T (XT X − zI)−1v(ℓ))
++ (XT X − zI)−1V Λ1/2(U T X(XT X − zI)−1v(ℓ))
++ (XT X − zI)−1V Λ1/2(U T U)Λ1/2(V T (XT X − zI)−1v(ℓ))
+=
+�
+λℓ⟨v(ℓ), (XT X − zI)−1v(ℓ)⟩(XT X − zI)−1XT u(ℓ)
++ λℓ⟨v(ℓ), (XT X − zI)−1v(ℓ)⟩(XT X − zI)−1v(ℓ) + θ2(ℓ)
+where ∥θ1(ℓ)∥, ∥θ2(ℓ)∥ = O≺(N −φ) since ∥U T U − I∥F ≺ N −φ.
+In particular, k extremal eigenvectors of T are a linear combination of (XT X−zI)−1XT u(ℓ) and (XT X−
+zI)−1v(ℓ).
+Suppose that aℓ(XT X−zI)−1XT u(ℓ)+bℓ(XT X−zI)−1v(ℓ) is an eigenvector of T with the corresponding
+eigenvalue −1. Thus, from Lemma B.4,
+−
+�
+aℓ(XT X − zI)−1XT u(ℓ) + bℓ(XT X − zI)−1v(ℓ)
+�
+= T
+�
+aℓ(XT X − zI)−1XT u(ℓ) + bℓ(XT X − zI)−1v(ℓ)
+�
+= −bℓ
+�
+λℓ
+�
+1
+zs(z) + 1
+�
+(XT X − zI)−1XT u(ℓ)
++ aℓ
+�
+λℓ(zs(z) + 1)(XT X − zI)−1v(ℓ) − bℓλℓ
+�
+1
+zs(z) + 1
+�
+(XT X − zI)−1v(ℓ) + �θ(ℓ)
+(B.10)
+for some �θ(ℓ), which is a linear combination of (XT X −zI)−1XT u(ℓ) and (XT X −zI)−1v(ℓ) with ∥�θ(ℓ)∥ =
+O≺(N −φ).
+Since U, V , and X are independent, (XT X − zI)−1XT u(ℓ) and (XT X − zI)−1v(ℓ) are linearly inde-
+37
+
+pendent with overwhelming probability. Thus, from (B.10),
+−aℓ = −bℓ
+�
+λℓ
+�
+1
+zs(z) + 1
+�
++ O≺(N −φ),
+−bℓ = aℓ
+�
+λℓ(zs(z) + 1) − bℓλℓ
+�
+1
+zs(z) + 1
+�
++ O≺(N −φ).
+It is then elementary to check that
+λℓ(zs(z) + 1) + 1 = O≺(N −φ),
+which has the solution
+z = (1 + λℓ)
+�
+1 + d0
+λℓ
+�
++ O≺(N −φ)
+if and only if λℓ > √d0. This proves the BBP result in our setting.
+We now turn to the proof of Theorem 3.4. To simplify the exposition, we focus on the case that SNRs are
+the same i.e., Λ = λI. For the spike prior in Assumption 3.1, suppose that a function q and its all derivatives
+are polynomially bounded in the sense of Assumption 3.1. Following the proof of Theorem 4.8 in [50], we
+define the error term from the local linear estimation of q(
+√
+NYij) by
+q(
+√
+NYij) = q(
+√
+NXij) +
+√
+λNuivT
+j q′(
+√
+NXij) + Rij
+where
+Rij = 1
+2q′′(
+√
+NXij + eij)λN(uivT
+j )2
+for some |eij| ≤ |
+√
+λNuivT
+j |. The Frobenius norm of R is bounded as
+∥R∥2
+F = Tr RT R = λ2N 2
+4
+M
+�
+i=1
+N
+�
+j=1
+(uivT
+j )4q′′(
+√
+NXij + eij)2
+≤ λ2N 2−4φ
+4
+M
+�
+i=1
+N
+�
+j=1
+(uivT
+j )2q′′(
+√
+NXij + eij)2.
+Since q′′ is polynomially bounded, q′′(
+√
+NXij + eij) is uniformly bounded by an N-independent constant.
+Thus, with overwhelming probability,
+∥R∥2 ≤ ∥R∥2
+F ≤ Cλ2N 2−4φ.
+Next, we approximate q(
+√
+NXij) by its mean. Let
+Eij = q′(
+√
+NXij) − E[q′(
+√
+NXij)],
+∆ij =
+√
+λNuivT
+j Eij.
+Then, ∥∆∥ ≺ N
+1
+2 −2φ∥E∥ and, since the entries of matrix E are i.i.d., centered and with finite moments, its
+norm ∥E∥ = O(
+√
+N) with overwhelming probability. (See, e.g., [18].) Thus, ∥∆∥ = O≺(N 1−2φ).
+Set
+Mq := E[q′(
+√
+NXij)],
+Vq := E[q(
+√
+NXij)2],
+�λ := λM 2
+q /Vq,
+38
+
+and
+Qij :=
+1
+�
+NVq
+q(
+√
+NXij).
+We have proved so far that the difference of the largest eigenvalue of Q + �λ
+1
+2 UV T and that of the matrix
+�
+1
+�
+NVq
+q(
+√
+NYij)
+�
+is O≺(N
+1
+2 −2φ), which is o(1) with overwhelming probability for φ > 1
+4. It is directly applicable to the case
+that Λ in our model with �Λℓℓ := λℓM 2
+q /Vq since the above process does not require any information of the
+SNRs. The BBP result holds the matrix Q+U �Λ
+1
+2 V T , which is another (additive) spiked rectangular matrix.
+This shows that the BBP result also holds for �Y with SNR matrix �Λ :=
+M 2
+q
+Vq Λ. This proves Theorem 3.4.
+B.4
+Proof of Theorem 3.5
+Recall that the spike prior satisfies the technical conditions in Assumption 3.1 with φ > 1/4. For the sake
+of brevity, we assume that Λ = λI. As in the additive case, we further assume that a function q and its all
+derivatives are polynomially bounded and consider the local linear approximation of q(
+√
+NYij),
+q(
+√
+NYij) = q(
+√
+NXij) + γ
+√
+NE[q′(
+√
+NXij)]
+�
+ℓ
+uiuT
+ℓ Xℓj + Rij + γ∆ij,
+(B.11)
+where
+Rij = 1
+2q′′�√
+NXij + θγ
+�
+ℓ
+uiuT
+ℓ
+√
+NXℓj
+� �
+γ
+�
+ℓ
+uiuT
+ℓ
+√
+NXℓj
+�2
+for some θ ∈ [−1, 1] and
+∆ij =
+√
+NEij
+�
+ℓ
+uiuT
+ℓ Xℓj,
+Eij = q′(
+√
+NXij) − E[q′(
+√
+NXij)].
+For any unit vectors a = (a1, a2, . . . , aM) and b = (b1, b2, . . . , bN),
+aT ∆b =
+�
+s
+�
+i,j
+aiui(s)Eijbj
+��
+ℓ
+uℓ(s)
+√
+NXℓj
+�
+=
+�
+s
+�
+i,j
+aiui(s)2bjEij
+√
+NXij +
+�
+s
+�
+i,j
+aiui(s)Eijbj
+�
+��
+ℓ̸=i
+uℓ(s)
+√
+NXℓj
+�
+�
+From the concentration inequalities such as Lemma B.5,
+�
+ℓ
+uiuT
+ℓ
+√
+NXℓj =
+�
+s
+�
+ℓ
+ui(s)uℓ(s)T √
+NXℓj ≺
+�
+s
+|ui(s)|
+��
+ℓ
+uℓ(s)2
+�1/2
+≺ N −φ.
+(B.12)
+Recall that ∥E∥ = O(
+√
+N) with overwhelming probability.
+Note that, by Assumption 3.1, the density
+function q have to be an odd function. Further, since q is an odd function (hence xq′(x) is an odd function
+39
+
+of x), the norm of the matrix whose (i, j)-entry is Eij
+√
+NXij is also O(
+√
+N). Thus,
+aT ∆b ≺ N −2φ + N
+1
+2 −φ,
+which shows that ∥∆∥ ≺ N
+1
+2 −φ. Moreover, since q′′ is polynomially bounded, following the proof of Theorem
+3.4 with (B.12),
+∥R∥2 ≤ ∥R∥2
+F ≤ CN 2−4φ.
+Thus, as in the additive case, the error terms Rij and ∆ij in (B.11) are negligible when finding the limit of
+the extreme eigenvalues of the transformed matrix.
+Set
+Mq := E[q′(
+√
+NXij)],
+Vq := E[q(
+√
+NXij)2],
+Eq = E[
+√
+NXijq(
+√
+NXij)],
+�γ := γMq/
+�
+Vq,
+and
+Qij :=
+1
+�
+NVq
+q(
+√
+NXij).
+With the approximation (B.11), we now focus on the largest eigenvalue of
+(Q + �γUU T X)T (Q + �γUU T X).
+Note that the assumption on the polynomial boundedness of q implies that the matrix Q is also a rectangular
+matrix satisfying the assumptions in Definition 2.2.
+Let G(z) and G(z) be the resolvents
+G ≡ G(z) := (QQT − zI)−1,
+G ≡ G(z) := (QT Q − zI)−1
+for z ∈ R outside an open interval containing [d−, d+]. We note that the following identities hold for G(z)
+and G(z):
+G(z)Q = QG(z),
+QT G(z)Q = I + zG(z).
+(B.13)
+As in the proof of Theorem 3.4, we consider
+(Q + �γUU T X)T (Q + �γUU T X) − zI
+= (QT Q − zI)(I + (QT Q − zI)−1(�γXT UU T Q + �γQT UU T X + �γ2XT UU T UU T X)).
+(B.14)
+Let
+L ≡ L(z) = G(z)(�γXT UU T Q + �γQT UU T X + �γ2XT UU T UU T X),
+Then, as in the proof of Theorem 3.4, if z is an eigenvalue of (Q + �γUU T X)T (Q + �γUU T X) (but not of
+QT Q), −1 is an eigenvalue of L(z). Again, the rank of L is at most 2k, with
+L · GQT U = �γGXT UU T QGQT U + �γGQT UU T XGQT U + �γ2GXT UU T UU T XGQT U,
+L · GXT U = �γGXT UU T QGXT U + �γGQT UU T XGXT U + �γ2GXT UU T UU T XGXT U,
+(B.15)
+and an eigenvector of L is a linear combination of GQT u(ℓ) and GXT u(ℓ) for 1 ≤ ℓ ≤ k.
+In the simplest case where Q is the identity mapping, Q = X, hence the rank of L is k, and the eigenvalue
+equation (B.15) is simplified to
+L · GQT U = �γGQT U(U T QGQT U) + �γGQT U(U T QGQT U) + �γ2GQT U(U T UU T QGQT U).
+(B.16)
+40
+
+In this case, GQT u(ℓ) are eigenvectors of L corresponding to the eigenvalue −1, i.e., L · GQT u(ℓ) =
+−GQT u(ℓ).
+The right side of (B.16) can be approximated as follows, which is a direct consequence of
+the isotropic local Marchenko–Pastur law (e.g., Theorem 2.5 of [17]).
+With the isotropic local Marchenko–Pastur law, (B.16) can be approximated by a deterministic vector
+equation on z (and s(z)), and the location of the k largest eigenvalues can be proved by solving the equation.
+In a general case where Q is not a multiple of X and the vectors GQT u(ℓ) and GXT u(ℓ) are linearly
+independent, however, the eigenvalue equation (B.15) contains other matrices U T QGQT U, U T QGXT U,
+and U T XGQT U, which cannot be estimated by Lemma B.4.
+For these matrices, we use the following
+lemma.
+Lemma B.6. Suppose that the assumptions in Lemma B.4 hold. Then,
+⟨u(ℓ1), XGQT u(ℓ2)⟩ = ⟨u(ℓ1), QGXT u(ℓ2)⟩ =
+�
+Eq
+�
+Vq
+(zs(z) + 1)
+�
+δℓ1ℓ2 + O≺(N −φ)
+and
+⟨u(ℓ1), XGXT u(ℓ2)⟩ =
+�
+E2
+q
+Vq
+zs(z)
+�
+d0s(z) + d0 − 1
+z
+�2
++ d0s(z) + d0 − 1
+z
+�
+δℓ1ℓ2 + O≺(N −φ).
+We defer the proof to Appendix B.6.
+With Lemma B.6, we are ready to finish the proof. From the definition of s(z) in Lemma B.4, we notice
+that
+s(z) =
+1
+1 − d0 − d0zs(z) − z ,
+(B.17)
+or
+z
+�
+d0s(z) + d0 − 1
+z
+�
+= − 1
+s(z) − z.
+(B.18)
+Set σ(z) := zs(z) + 1. By applying Lemmas B.4 and B.6 to (B.16), for 1 ≤ ℓ ≤ k
+L · GQT u(ℓ) = �γ⟨u(ℓ), QGQT u(ℓ)⟩ · GXT u(ℓ) + �γ⟨u(ℓ), XGQT u(ℓ)⟩ · GQT u(ℓ)
++ ∥u(ℓ)∥2�γ2⟨u(ℓ), XGQT u(ℓ)⟩ · GXT u(ℓ)
+= �γσ(z)GXT u(ℓ) + �γσ(z) Eq
+�
+Vq
+GQT u(ℓ) + �γ2 Eq
+�
+Vq
+σ(z)GXT u(ℓ) + θ1(ℓ) ,
+(B.19)
+and
+L · GXT u(ℓ) = �γ⟨u(ℓ), QGXT u(ℓ)⟩ · GXT u(ℓ) + �γ⟨u(ℓ), XGXT u(ℓ)⟩ · GQT u(ℓ)
++ ∥u(ℓ)∥2�γ2⟨u(ℓ), XGXT u(ℓ)⟩ · GXT u(ℓ)
+= �γσ(z) Eq
+�
+Vq
+GXT u(ℓ) + �γ
+��
+σ(z) +
+σ(z)
+σ(z) − 1
+� E2
+q
+Vq
+−
+σ(z)
+σ(z) − 1
+�
+GQT u(ℓ)
++ �γ2
+��
+σ(z) +
+σ(z)
+σ(z) − 1
+� E2
+q
+Vq
+−
+σ(z)
+σ(z) − 1
+�
+GXT u(ℓ) + θ2(ℓ) ,
+(B.20)
+for some θ1(ℓ), θ2(ℓ), which are linear combinations of GQT u(ℓ) and GXT u(ℓ), with ∥θ1(ℓ)∥, ∥θ2(ℓ)∥ =
+O≺(N −φ).
+Suppose that aℓGQT u(ℓ)+bℓGXT u(ℓ) is an eigenvector of L with the corresponding eigenvalue −1. From
+41
+
+(B.19), (B.20), and the linear independence between GQT u(ℓ) and GXT u(ℓ), we find the relation
+−aℓ = aℓ�γσ(z) Eq
+�
+Vq
++ bℓ�γσ(z)2
+σ(z) − 1 · E2
+q
+Vq
+− bℓ�γσ(z)
+σ(z) − 1 + O(N −φ),
+−bℓ = aℓ�γσ(z) + aℓ�γ2σ(z) Eq
+�
+Vq
++ bℓ�γσ(z) Eq
+�
+Vq
++ bℓ�γ2σ(z)2
+σ(z) − 1 · E2
+q
+Vq
+− bℓ�γ2σ(z)
+σ(z) − 1 + O(N −φ).
+We then find that
+bℓ
+aℓ
+�
+1 + �γσ(z) Eq
+�
+Vq
+�
++ �γσ(z) − �γ = O(N −φ)
+and
+aℓ
+�
+1 + �γσ(z) Eq
+�
+Vq
+�
+= bℓ
+�
+�γσ(z)
+σ(z) − 1
+�
+1 − σ(z) · E2
+q
+Vq
+��
++ O(N −φ),
+which implies that
+1 + 2�γσ(z) Eq
+�
+Vq
++ �γ2σ(z) = 1 +
+�
+2γMqEq + γ2M 2
+q
+Vq
+�
+σ(z) = O(N −φ).
+(B.21)
+From the explicit formula for s, it is not hard to check that (B.21) holds if and only if
+λq := 2γMqEq + γ2M 2
+q
+Vq
+>
+�
+d0
+and
+z = (1 + λq)
+�
+1 + d0
+λq
+�
++ O(N −φ).
+(B.22)
+We see that it is valid for general Λ in our model, since the above process also does not require any information
+of the SNRs as in the additive case. Now, the desired theorem follows from the direct computation for the
+case q = hαg; see also Appendix B.5.2.
+B.5
+Optimal entrywise transformation
+B.5.1
+Additive model
+Recall that
+E[q′(
+√
+NWij)] = E[q′(
+√
+NXij)] = Mq,
+E[q(
+√
+NWij)2] = E[q(
+√
+NXij)2] = Vq.
+Following the proof of Theorem 3.4 in Appendix B.3, it is not hard to see that the effective SNR is maximized
+by optimizing M 2
+q /Vq. Such an optimization problem was already considered in [50] for the spiked Wigner
+matrix. For the sake of completeness, we solve this problem by using the calculus of variations. Recall the
+density of random variables
+√
+NWij and
+√
+NXij is g.
+To optimize q, we need to maximize
+�� ∞
+−∞
+q′(x)g(x)dx
+�2
+/
+�� ∞
+−∞
+q(x)2g(x)dx
+�
+=
+�� ∞
+−∞
+q(x)g′(x)dx
+�2
+/
+�� ∞
+−∞
+q(x)2g(x)dx
+�
+.
+(B.23)
+Putting (q + εη) in place of q in (B.23) and differentiating with respect to ε, we find that the optimal q
+42
+
+satisfies
+�� ∞
+−∞
+η(x)g′(x)dx
+� �� ∞
+−∞
+q(x)2g(x)dx
+�
+=
+�� ∞
+−∞
+q(x)η(x)g(x)dx
+� �� ∞
+−∞
+q(x)g′(x)dx
+�
+(B.24)
+for any η. It is then easy to check that q = −Cg′/g is the only solution of (B.24). Since the value in (B.23)
+does not change if we replace q by Cq, and the effective SNR is increased with the entrywise transform −g′/g
+is the optimal entrywise transformation for PCA.
+B.5.2
+Multiplicative model
+As we can see from the proof of Theorem 3.5 in Appendix B.4, we need to maximize
+2
+�� ∞
+−∞ xq(x)g(x)dx
+� �� ∞
+−∞ q′(x)g(x)dx
+�
++ γ
+�� ∞
+−∞ q′(x)g(x)dx
+�2
+�� ∞
+−∞ q(x)2g(x)dx
+�
+=
+−2
+�� ∞
+−∞ xq(x)g(x)dx
+� �� ∞
+−∞ q(x)g′(x)dx
+�
++ γ
+�� ∞
+−∞ q(x)g′(x)dx
+�2
+�� ∞
+−∞ q(x)2g(x)dx
+�
+.
+(B.25)
+Putting (q + εη) in place of q in (B.23) and differentiating with respect to ε, we find that the optimal q
+satisfies
+− 2
+��
+xηg
+� ��
+qg′
+� ��
+q2g
+�
+− 2
+��
+xqg
+� ��
+ηg′
+� ��
+q2g
+�
++ 2γ
+��
+ηg′
+� ��
+qg′
+� ��
+q2g
+�
++ 4
+��
+qηg
+� ��
+xqg
+� ��
+qg′
+�
+− 2γ
+��
+qg′
+�2 ��
+qηg
+�
+= 0
+(B.26)
+which is written with slight abuse of notation such as
+�
+xηg =
+� ∞
+−∞ xη(x)g(x)dx. Since the equation contains
+the terms
+�
+xηg,
+�
+ηg′,
+�
+qηg,
+it is natural to consider an ansatz
+q(x) = −g′(x)
+g(x) + αx
+(B.27)
+for a constant α. Collecting the terms involving
+�
+xηg and the terms involving
+�
+ηg′, we get
+2(Fg + α)(Fg + 2α + α2) − 4α(1 + α)(Fg + α) − 2αγ(Fg + α)2 = 0
+and
+−2(1 + α)(Fg + 2α + α2) − 2γ(Fg + α)(Fg + 2α + α2) + 4(1 + α)(Fg + α) + 2γ(Fg + α)2 = 0.
+We can then check that
+α = αg =
+−γFg +
+�
+4Fg + 4γFg + γ2F 2g
+2(1 + γ)
+,
+43
+
+and hence (B.26) is satisfied with
+q(x) = −g′(x)
+g(x) +
+−γFg +
+�
+4Fg + 4γFg + γ2F 2g
+2(1 + γ)
+x.
+The corresponding effective SNR
+λhαg ≡ λg = γ + γ2Fg
+2
++
+γ
+�
+4Fg + 4γFg + γ2F 2g
+2
+.
+For a general α, when the entrywise transform hα is applied, the effective SNR
+λhα = 2γ(1 + α)(Fg + α) + γ2(α + Fg)2
+α2 + 2α + Fg
+,
+In particular, if α =
+�
+Fg,
+λh√
+Fg = γ(1 +
+�
+Fg) + γ2
+2 (Fg +
+�
+Fg) ≥ 2γ + γ2 = λ
+where the inequality is strict if Fg > 1.
+B.6
+Proof of Lemma B.6
+B.6.1
+Key ingredient: Entrywise local estimates
+Recall the definition of the random matrices X and Q. The couple of random matrices (X, Q) is one example
+of the following concept for a coupled random matrices:
+Definition B.7 (Entrywise correlated random matrices). Suppose that A and B are M × N random rectan-
+gular matrices in Definition 2.2 satisfying the following conditions:
+• For all 1 ≤ a, b ≤ M and 1 ≤ α, β ≤ N, Aaα and Bbβ are dependent only when a = b and α = β.
+• For all a, α, E[Aaα] = E[Baα] = 0, NE[A2
+aα] = wA, NE[B2
+aα] = wB, and NE[AaαBaα] = wAB.
+• For any positive integer p, there exists Cp, independent of N, such that
+N
+p
+2 E[Ap
+aα], N
+p
+2 E[Bp
+aα] ≤ Cp
+for all a, α.
+A couple of random matrices (A, B) is called the entrywise correlated.
+The key estimates in the proof of Lemma B.6 are the exact bounds on the entries of K := Q(QQT −
+zI)−1XT and K := X(QQT − zI)−1XT . We prove the following lemma for the entrywise correlated random
+matrices (A, B), which exactly contains the desired result.
+Lemma B.8. Let (A, B) be the entrywise correlated random matrices with wB = 1. For z ∈ R outside an
+open interval containing [d−, d+],
+|(A(BT B − zI)−1AT )ij − (wAs(z) + w2
+ABzs(z)s(z)2)δij| = O≺(N −1/2),
+(B.28)
+44
+
+|(A(BT B − zI)−1BT )ij − (wABs(z) + wABzs(z)s(z)2)δij| = O≺(N −1/2)
+(B.29)
+and
+|(B(BT B − zI)−1AT )ij − (wABs(z) + wABzs(z)s(z)2)δij| = O≺(N −1/2).
+(B.30)
+Remark B.9. Recall that σ(z) = zs(z) + 1. For (X, Q), since wX = wQ = 1 and wXQ = Eq/
+�
+Vq, we have
+the following:
+For z ∈ R outside an open interval containing [d−, d+],
+|Kij − �s(z)δij| = O≺(N −1/2),
+|Kij − ˇs(z)δij| = O≺(N −1/2),
+(B.31)
+where
+�s(z) := σ(z) Eq
+�
+Vq
+,
+ˇs(z) := zs(z)
+�
+d0s(z) + d0 − 1
+z
+�2 E2
+q
+Vq
++
+�
+d0s(z) + d0 − 1
+z
+�
+.
+(B.32)
+B.6.2
+Linearization
+We consider
+G ≡ GB(z) = (BBT − zI)−1,
+G ≡ GB(z) = (BT B − zI)−1.
+In the proof of Lemma B.8, we use the formalism known as the linearization to simplify the computation.
+We define an (M + N) × (M + N) symmetric matrix HB by
+HB ≡ HB(z) =
+�−zIM
+B
+BT
+−IN
+�
+,
+(B.33)
+where IM and IN are the identity matrices with size M and N, respectively.
+Let RB(z) = HB(z)−1.
+(For the invertibility of HB(z), we refer to Section 5.1 in [38].)
+By Schur’s
+complement formula,
+RQ(z) =
+� GB(z)
+GB(z)B
+BT GB(z)
+zGB(z)
+�
+.
+(B.34)
+Therefore,
+Rab(z) = (BBT − zI)−1
+ab = Gab(z),
+Rαβ(z) = z(BT B − zI)−1
+α−M,β−M = zGα−M,β−M(z),
+(B.35)
+and
+Rαa(z) = Raα(z) = (GB)a,α−M(z),
+(B.36)
+where we use lowercase Latin letters a, b, c, . . . for indices from 1 to M and Greek letters α, β, γ, . . . for
+indices from (M + 1) to (M + N). We also use uppercase Latin letters A, B, C, . . . for indices from 1 to
+(M + N). In the rest of Appendix B, we omit the subscript Q for brevity.
+For T ⊂ {1, 2, . . . , M + N}, we define the matrix minor H(T) by
+(H(T))AB := 1{A,B /∈T}HAB .
+(B.37)
+Moreover, for A, B /∈ T we define
+R(T)
+AB(z) := (H(T))−1
+AB,
+(B.38)
+In the definitions above, we abbreviate ({A}) by (A); similarly, we write (AB) instead of ({A, B}).
+We have the following resolvent (decoupling) identities for the matrix entries of R and R(T), which are
+45
+
+elementary consequences of Schur’s complement formula; see e.g. Lemma 5.1 of [38].
+Lemma B.10 (Resolvent identities for R). Suppose that z ∈ R is outside an open interval containing
+[d−, d+].
+- For a ̸= b,
+Rab = −Raa
+�
+α
+HaαR(a)
+αb = −Rbb
+�
+β
+R(b)
+aβHβb.
+- For α ̸= β,
+Rαβ = −Rαα
+�
+a
+HαaR(α)
+aβ = −Rββ
+�
+b
+R(β)
+αb Hbβ.
+- For any a and α,
+Raα = −Raa
+�
+β
+HaβR(a)
+βα = −Rαα
+�
+b
+R(α)
+ab Hbα.
+- For A, B ̸= C,
+RAB = R(C)
+AB + RACRCB
+RCC
+.
+Throughout this section, we will frequently use the estimate that all entries of X and Q (and hence all
+off-diagonal entries of W) are O≺(N −1/2), which holds since all moments of the entries of
+√
+NQ and
+√
+NX
+are bounded. For the entries of R, we have the following estimates:
+Lemma B.11. Let
+s(z) =
+�
+d0s(z) + d0 − 1
+z
+�
+.
+(B.39)
+For z ∈ R outside an open interval containing [d−, d+],
+|Rij(z) − s(z)δij| , |Rµν(z) − zs(z)δµν| , |Riµ(z)| ≺ N −1/2.
+(B.40)
+Proof of Lemma B.11. The first two estimates can be checked from Theorem 2.5 (and Remark 2.7) in [17]
+with the deterministic unit vectors v = ei and w = ej where ei ∈ RN or RM is a standard basis vector
+whose i-th coordinate is 1 and all other coordinates are zero. For the last estimate, we apply Lemma B.10
+to find that
+Riµ(z) = −Rii
+�
+α
+HiαR(i)
+αµ.
+Since Hiα and R(i)
+αµ are independent, R(i)
+αµ ≺ N −1/2 for α ̸= µ, and R(i)
+µµ = Θ(1) with overwhelming probability,
+we find from Lemma B.5 that
+�
+α
+HiαR(i)
+αµ ≺
+�
+1
+N
+�
+α
+|R(i)
+αµ|2
+�1/2
+≺ N −1/2.
+Proof of Lemma B.8. Throughout this section, for the sake of brevity, we will use the notation
+Baα := Ba,(α−M) = Haα,
+Aaα := Aa,(α−M).
+We begin by estimating the diagonal entry (BGAT )ii. From Schur’s complement formula, (B.35), we can
+decompose it into
+(BGAT )ii = 1
+z
+�
+α
+HiαRααAiα + 1
+z
+�
+α̸=β
+HiαRαβAiβ.
+(B.41)
+46
+
+From concentration inequalities it is not hard to see that
+�
+α
+BiαAiα = E[BiαAiα] + O≺(N −1/2) = wAB + O≺(N −1/2).
+Applying Lemma B.11, we find for the first term in the right side of (B.41) that
+1
+z
+�
+α
+HiαRααAiα = wABs(z) + O≺(N −1/2).
+(B.42)
+We next estimate the second term in the right side of (B.41). We expand it with the resolvent identities
+in Lemma B.10 as follows:
+�
+α̸=β
+HiαRαβAiβ =
+�
+α̸=β
+HiαR(i)
+αβAiβ +
+�
+α̸=β
+Hiα
+RαiRiβ
+Rii
+Aiβ
+=
+�
+α̸=β
+HiαR(i)
+αβAiβ +
+�
+α̸=β
+Hiα
+RαiRiβ
+s(z)
+Aiβ + O≺(N −1/2).
+(B.43)
+Here, in the estimate for the second term, we simply counted the power (of N) as it involves two indices for
+the sum (hence O(N 2) terms) of Hiα, Rαi, Riβ, Aiβ ≺ N −1/2, hence �
+α̸=β HiαRαiRiβAiβ = O≺(1). Applying
+Lemma B.5 to the first term in the right side of (B.43),
+�
+α̸=β
+HiαR(i)
+αβAiβ ≺
+�
+� 1
+N 2
+�
+α,β
+|R(i)
+αβ|2
+�
+�
+1/2
+≺ N −1/2.
+For the second term in the right side of (B.43), we further expand it to find
+�
+α̸=β
+HiαRαiRiβAiβ =
+�
+α̸=β
+HiαRαiRiβAiβ = −
+�
+α̸=β
+Hiα
+�
+Rii
+�
+µ
+R(i)
+αµHµiRiβAiβ
+�
+Note that
+�
+µ
+R(i)
+αµHµi ≺ N −1/2,
+as in the proof of Lemma B.11. Since
+|Rij − s(z)| ≺ N −1/2,
+Riβ = R(α)
+iβ + RiαRαβ
+Rαα
+= R(α)
+iβ + N −1,
+we have
+−
+�
+α̸=β
+Hiα
+�
+Rii
+�
+µ
+R(i)
+αµHµiRiβAiβ
+�
+= −s(z)
+�
+α̸=β
+Hiα
+��
+µ
+R(i)
+αµHµiR(α)
+iβ Aiβ
+�
++ O≺(N −1/2)
+= −s(z)
+�
+α̸=β
+Hiα
+�
+� �
+µ:µ̸=α
+R(i)
+αµHµiR(α)
+iβ Aiβ
+�
+� − s(z)
+�
+α̸=β
+(Hiα)2R(i)
+ααR(α)
+iβ Aiβ + O≺(N −1/2).
+(B.44)
+47
+
+Applying Lemma B.5 again to the first term in the right side of (B.44),
+�
+α̸=β
+Hiα
+�
+� �
+µ:µ̸=α
+R(i)
+αµHµiR(α)
+iβ Aiβ
+�
+� ≺
+�
+�
+� 1
+N
+�
+α
+������
+�
+β:β̸=α
+�
+� �
+µ:µ̸=α
+R(i)
+αµHµi
+�
+� R(α)
+iβ Aiβ
+������
+2�
+�
+�
+1/2
+≺
+�
+�
+� 1
+N
+�
+α
+�
+� �
+β:β̸=α
+N −1/2 ���R(α)
+iβ Aiβ
+���
+�
+�
+2�
+�
+�
+1/2
+≺ N −1/2.
+Similarly, by expanding R(α)
+iβ , we find for the second term in the right side of (B.44) that
+−s(z)
+�
+α̸=β
+(Hiα)2R(i)
+ααR(α)
+iβ Aiβ = zs(z)s(z)
+�
+α̸=β
+(Hiα)2R(α)
+ii
+�
+ν:ν̸=α
+H(α)
+iν R(iα)
+νβ Aiβ + O≺(N −1/2)
+= zs(z)2s(z)
+�
+α̸=β
+(Hiα)2
+�
+ν:ν̸=α,β
+HiνR(iα)
+νβ Aiβ + zs(z)2s(z)
+�
+α̸=β
+(Hiα)2HiβR(iα)
+ββ Aiβ + O≺(N −1/2)
+= z2s(z)2s(z)2 �
+α̸=β
+(Hiα)2HiβAiβ + O≺(N −1/2),
+where we used Lemma B.5 to find
+�
+ν̸=β:ν,β̸=α
+HiνR(iα)
+νβ Aiβ ≺
+�
+� 1
+N 2
+�
+ν̸=β:ν,β̸=α
+���R(iα)
+νβ
+���
+2
+�
+�
+1/2
+≺ N −1/2.
+Thus, since wB = 1,
+�
+α̸=β
+HiαRαiRiβAiβ = z2s(z)2s(z)2 �
+α̸=β
+(Hiα)2HiβAiβ + O≺(N −1/2)
+= z2s(z)2s(z)2wAB + O≺(N −1/2),
+and putting it back to (B.43) and (B.41), together with (B.42), we conclude that
+(AGB)ii = wABs(z) + wABzs(z)s(z)2 + O≺(N −1/2) = wABσ(z) Eq
+�
+Vq
++ O≺(N −1/2),
+(B.45)
+where we used the identity zs(z)s(z) = −σ(z). In the same manner, we also find that
+(AGA)ii = 1
+z
+�
+α
+AiαRααAiα + zs(z)s(z)2 �
+α̸=β
+AiαHiαHiβAiβ + O≺(N −1/2)
+= wAs(z) + w2
+ABzs(z)s(z)2 + O≺(N −1/2).
+(B.46)
+We next estimate the off-diagonal entry (AGB)ij. We expand it as
+(AGB)ij = 1
+z
+�
+α,β
+HiαRαβAjβ = 1
+z
+�
+α,β
+HiαR(i)
+αβAjβ + 1
+z
+�
+α,β
+Hiα
+RαiRiβ
+Rii
+Ajβ
+= 1
+z
+�
+α,β
+HiαR(ij)
+αβ Ajβ + 1
+z
+�
+α,β
+Hiα
+R(i)
+αjR(i)
+jβ
+R(i)
+jj
+Ajβ + 1
+z
+�
+α,β
+Hiα
+R(j)
+αi R(j)
+iβ
+R(i)
+jj
+Ajβ + O≺(N −1/2)
+(B.47)
+48
+
+From Lemma B.5,
+�
+α,β
+HiαR(ij)
+αβ Ajβ ≺ N −1/2.
+We also have
+�
+α,β
+Hiα
+R(i)
+αjR(i)
+jβ
+R(i)
+jj
+Ajβ ≺
+�
+�
+� 1
+N
+�
+α
+������
+�
+β
+R(i)
+αjR(i)
+jβ
+R(i)
+jj
+Ajβ
+������
+2�
+�
+�
+1/2
+≺
+�
+�
+� 1
+N
+�
+α
+������
+�
+β
+N −3/2
+������
+2�
+�
+�
+1/2
+≺ N −1/2
+and a similar estimate holds for the third term in the right side of (B.47). Thus,
+(AGB)ij ≺ N −1/2
+In the same manner, we also find that (AGA)ij ≺ N −1/2. Together with (B.45) and (B.46), this proves
+Lemma B.8.
+B.6.3
+Isotropic local law
+We also assume that wB = 1 and use the same notation in previous section. Then our goal is to prove the
+following statement:
+Lemma B.12. Let (A, B) be the entrywise correlated random matrices where wB = 1 and x, y are determin-
+istic and ℓ2 - normalized vectors in RM. Then, for z ∈ R outside an open interval containing [d−, d+],
+⟨x, A(BT B − zI)−1AT y⟩ = (wAs(z) + w2
+ABzs(z)s(z)2)⟨x, y⟩ + O≺(N −1/2).
+Proof of Lemma B.12. Note that, due to polarization identity, we suffice to prove for ⟨x, A(BT B−zI)−1AT x⟩.
+Recall that we have
+(A(BT B − zI)−1AT )ij = (wAs(z) + w2
+ABzs(z)s(z)2)δij + O≺(N −1/2)
+and
+(A(BT B − zI)−1BT )ij = (B(BT B − zI)−1AT )ij = (wABs(z) + wABzs(z)s(z)2)δij + O≺(N −1/2).
+Once the entrywise local law is given, the proof of the isotropic (or anisotropic) type law follows exactly as
+in [17]. To be more precisely, we can write
+⟨x, A(BT B − zI)−1AT x⟩ =
+�
+i
+xi(AGAT )iixi +
+�
+i̸=j
+xi(AGAT )ijxj.
+Then the entrywise local law implies
+�
+i
+xi(AGAT )iixi − (wAs(z) + w2
+ABzs(z)s(z)2)⟨x, x⟩
+=
+�
+i
+x2
+i
+�
+(AGAT )ii − (wAs(z) + w2
+ABzs(z)s(z)2)
+�
+≺ N −1/2,
+49
+
+and so the main difficulty is to control the off-diagonal part
+ZAB :=
+�
+α,β
+�
+i̸=j
+xiAiαGαβAjβxj = O≺(N −1/2).
+For instance, for the sample covariance matrix case
+⟨x, BGBT x⟩ = (zs(z) + 1)⟨x, x⟩ + O≺(N −1/2)
+= (s(z) + zs(z)s(z)2)⟨x, x⟩ + O≺(N −1/2)
+was proved in [17] by proving the following bound for higher moments
+E|ZB|p ≺ N −p/2
+(B.48)
+for any large and even p, where
+ZB :=
+�
+i̸=j
+xi(BGBT )ijxj = z
+�
+i̸=j
+xiGijxj.
+In particular, the (B.48) have proved by using the standard maximal expansion method in [17] and [2], which
+only requires the independence between each element, the boundedness of the moment of each entries, and
+the entrywise local law. Thus, from the definition of the entrywise correlated random matrices (A, B), it can
+be expected that
+E|ZAB|p ≺ N −p/2
+also holds for any large and even p, by expanding maximally Gαβ instead of Gab as in (B.47). Then, we can
+conclude the proof by using Markov inequality.
+To prove such an argument , we only need to check what is changing. First, we express the p-th moment
+of ZAB by
+E|ZAB|p = E
+�
+b11̸=b12
+· · ·
+�
+bp1̸=bp2
+�
+�
+p/2
+�
+k=1
+xbk1(AGAT )bk1bk2xbk2
+�
+�
+�
+�
+p
+�
+k=p/2+1
+xbk1(AGAT )bk1bk2xbk2
+�
+� .
+(B.49)
+Let T = {bk1} ∪ {bk2} be the set of indices of x appearing in the fixed summand of the representation of
+the p-th moment of ZAB. Then our goal is to decompose the off-diagonal entry of the matrix (AGAT ) into the
+two parts by using Lemma B.10, where one consists of the finite number of the maximally expanded term
+and the other consists of the terms containing a sufficiently large number of off-diagonal entries. We note
+that the latter case is small enough due to the entrywise local laws of off-diagonal entries, and so the leading
+order term contained in the formal.
+Step 1 : The maximal expansion for the off-diagonal entries of (AGBA).
+In our case, the maximally expanded terms (cf. Definition 5.4 of [17]) refer to terms that have one of the
+following forms: (AG(T\a,b)AT )ab, (AG(T\a,b)BT )ab, (BG(T\a,b)AT )ab or (BG(T\a,b)BT )ab = z(G(T\a,b))ab, for some
+a ̸= b ∈ T. To proceed, we use the following operation successively :
+Operation (a)
+Let T ⊂ {1, . . . M} be a set of indices.
+50
+
+• For a ̸= b and c /∈ T ,
+(AG(T )AT )ab = (AG(T c)AT )ab + (AG(T )BT )ac(BG(T )AT )cb
+z(G(T ))cc
+• For a ̸= b and c /∈ T ,
+(AG(T )BT )ab = (AG(T c)BT )ab + (AG(T )BT )ac(BG(T )BT )cb
+z(G(T ))cc
+= (AG(T c)BT )ab + (AG(T )BT )ac(G(T ))cb
+(G(T ))cc
+• For a ̸= b /∈ T and a, b ̸= c
+1
+z (BG(T )BT )ab = (G(T ))ab = (G(T c))ab + (G(T ))ac(G(T ))cb
+(G(T ))cc
+• For a ̸= b /∈ T
+1
+(G(T ))aa
+=
+1
+(G(T b))aa
+−
+(G(T ))ab(G(T ))ba
+(G(T ))aa(G(T b))aa(G(T ))bb
+We then observe that the expanded terms contains at most two crossed terms (AG(T\a,b)BT )ab, (BG(T\a,b)AT )ab
+and each expansions produce two types of terms, the first one has one more additional upper index, and
+the second one at least one more additional off-diagonal entry of BGAT , AGBT or BGBT . Moreover, we also
+remark that the denominators are always the diagonal entries of the resolvent G(T ).
+It can be seen that the above expansion formulas eventually play the same role as operation (a) in [17].
+Therefore, to obtain the desired decomposition, we only need to iterate the operation (a) until it can no
+longer be expanded or contains sufficiently many off-diagonal entries.
+Step 2 : The further expansions for the maximally expanded off-diagonal entries
+We further expand the maximally expanded term by using the following operations :
+Operations (b) (and (c))
+• For a ̸= b ∈ T
+(G(T\a,b))ab = z(G(T\a,b))aa(G(T\b))bb(BG(T)BT )ab.
+• Furthermore, we use the following type expansion, which is from the above formula, to the terms
+(G(T\a,b))aa and (G(T\a,b))bb
+(G(T\a,b))aa = (G(T\a))aa + (G(T\a,b))ab(G(T\a,b))ba
+(G(T\a,b))bb
+= (G(T\a))aa + z2(G(T\a,b))aa(G(T\a))aa(G(T\b))bb(BG(T)BT )2
+ab.
+Then, this expansion splits such not-maximally expanded term into two parts, one is maximally ex-
+panded and the other is a monomial expressed as the product of itself, the diagonal entry, and the
+maximally expanded terms. In particular, it can be seen that the number of the off-diagonal entries
+included in the latter monomial increases by exactly two.
+51
+
+• For a ̸= b ∈ T
+(AG(T\a,b)AT )ab = (AG(T)AT )ab + z(G(T\b))bb(AG(T)BT )ab(BG(T)AT )bb
++ (AG(T\a,b)BT )aa(BG(T\a,b)AT )ab
+z(G(T\a,b))aa
+= (AG(T)AT )ab + z(G(T\b))bb(AG(T)BT )ab(BG(T)AT )bb
++ z(G(T\a,b))aa(AG(T)BT )aa
+�
+(BG(T)AT )ab + z(G(T\b))bb(BG(T)BT )ab
+�
++ z2(G(T\a,b))aa(G(T\b))bb(AG(T)BT )ab(BG(T)BT )ab×
+�
+(BG(T)AT )ab + z(G(T\b))bb(BG(T)BT )ab
+�
+.
+since
+(AG(T\a,b)BT )aa = −z(G(T\a,b))aa(AG(T\b)BT )aa
+= −z(G(T\a,b))aa
+�
+(AG(T)BT )aa + z(G(T\b))bb(AG(T)BT )ab(BG(T)BT )ab
+�
+and
+(BG(T\a,b)AT )ab = −z(G(T\a,b))aa
+�
+(BG(T)AT )ab + z(G(T\b))bb(BG(T)BT )ab
+�
+.
+The expansion of the first two monomials terminated since every term were maximally expanded. After
+this, for any fixed positive integer ℓ, we expand the term which contains the term (G(T\a,b))aa until
+the last term is a monomial containing ℓ or more off-diagonal entries by applying the first formula
+recursively to the not-maximally expanded diagonal entry (G(T\a,b))aa.
+• For a ̸= b ∈ T
+(AG(T\a,b)BT )ab = −z(G(T\b))bb(AG(T)BT )ab + (AG(T\a,b)BT )aa(G(T\a,b))ab
+(G(T\a,b))aa
+= −z(G(T\b))bb(AG(T)BT )ab
+− z2(G(T\a,b))aa(AG(T)BT )aa(G(T\b))bb(BG(T)BT )ab
+− z3(G(T\a,b))aa(G(T\b))2
+bb(AG(T)BT )ab(BG(T)BT )ab(BG(T)BT )ab
+Even in this case, we also expand the second and third monomials recursively by applying the first
+formula to not-maximally expanded diagonal entry (G(T\a,b))aa.
+In particular, we have two observations from the above operations.
+• The expansions of the maximally expanded off-diagonal entry consist of the monomials containing only
+an odd number of off-diagonal entries: (BG(T)AT )ab, (AG(T)BT )ab and (BG(T)BT )ab.
+• The diagonal entries (AG(T)BT )aa = (BG(T)AT )aa for a ∈ T, appear in the expanded term by implement-
+ing operation (b) and (c). These terms can be interpreted as a loop of the vertex a in the structure of
+the graph considered in [17], since like the maximally expended diagonal entry, these terms are com-
+parable to wABs(z) by the entrywise local law. Therefore, similar to the maximally expanded diagonal
+entry, terms of such types have no effect on the partial expectation techniques in subsection 5.13 of
+[17]. This part will be explained in more detail in the next step.
+As with the previous step, from the explanations depicted in each expansion formula, we can see that
+the above expansions eventually play the same role as operations (b) and (c) in [17].
+52
+
+Step 3 : The further expansions for the maximally expanded diagonal entries
+Finally, unless we end up with an expression that includes a sufficiently large numbers of off-diagonal resolvent
+entries (such trivial leaves are dealt with separately in Subsection 5.11 of [17]), we need to expand the
+maximally expanded diagonal elements (AG(T)BT )aa = (BG(T)AT )aa and (G(T\a))aa for a ∈ T appearing in
+the non-trivial leaves (cf. Subsection 5.12 ∼ 14 of [17]), where we need to slightly adjust the proof to the
+setting. These terms corresponds to the maximally expanded diagonal G-edge in [17].
+First, for c ∈ T,
+1
+(G(T\c)
+B
+)cc
+= −z − z(BG(T)
+B
+BT )cc.
+(B.50)
+We note that |(G(T))µµ − s(z)| ≺ N −1/2 by following the proof of the entrywise local law. Using (B.50) and
+the facts zs(z)s(z) = −(zs(z) + 1) and |s(z)| ≍ 1, we see that
+1
+(G(T\c))cc
+=
+1
+s(z) − z
+�
+(BG(T)BT )cc − s(z)
+�
+and this implies that
+(G(T\c))cc =
+ℓ−1
+�
+k=0
+(s(z))k+1zk �
+(BG(T)BT )cc − s(z)
+�k
++ O≺(N −ℓ/2)
+for any integer ℓ ≥ 1 since (BG(T)BT )cc − s(z) is O≺(N −1/2), by using Lemma B.5.
+Similarly, for a ∈ T, we see that
+1
+(AG(T)BT )aa
+=
+1
+wABs(z) − (AG(T)
+B
+BT )aa − wABs(z)
+wABs(z)(AG(T)
+B
+BT )aa
+(B.51)
+and so
+(AG(T)BT )aa = wABs(z) − (AG(T)BT )aa
+wABs(z)−(AG(T)BT )aa
+wABs(z)
+1 − wABs(z)−(AG(T)BT )aa
+wABs(z)
+.
+By using the estimate
+(AG(T)BT )aa − wABs(z) =
+�
+µ̸=ν
+Aaµ(G(T))µνBaν +
+�
+µ
+AaµBaµ
+�
+(G(T))µµ − s(z)
+�
++ s(z)
+�
+1
+N
+�
+µ
+(NAaµBaµ − wAB)
+�
+≺ N −1/2,
+we have the following series expansion for any integers ℓ ≥ 1,
+(AG(T)BT )aa = wABs(z) − (AG(T)BT )aa1(ℓ ≥ 2)
+ℓ−1
+�
+k=1
+(wABs(z))−k �
+wABs(z) − (AG(T)BT )aa
+�k
++ O≺(N −ℓ/2)
+which corresponds to the term (5.42) in [17].
+This way we end up with an expression where only contains the resolvent terms of the type (AG(T)AT )ab,
+(AG(T)BT )ab, (BG(T)AT )ab or (BG(T)BT )ab = (G(T))ab, for some a ̸= b ∈ T. In other words, the x indices
+and the indices of the resolvent entries are completely decoupled; only explicit products of entries of (A, B)
+53
+
+represent the connections between them.
+Step 4 : Sketch of the rest of the proof.
+Through previous steps, for our case (AGAT ), we observed the modified version of the operations, which are
+done for the resolvents G and G in [17].
+After with these modifications, it can be seen that the rest procedures (Step 6 ∼ 8 in [17]) of the proof for
+the non-trivial leaves with the stopping rule, which relies on the number of off-diagonal terms (cf. Definition
+5.7 of [17]), are also valid for the ZAB.
+More precisely, by using the entrywise laws and H¨older’s inequality, the same estimation also holds for
+the trivial leave as in Subsection 5.11. Furthermore, the most of the finitely generated non-trivial leaves have
+a decay N −p/2 also by applying the same argument in the case of the trivial leaves (Subsection 5.12 in [17]),
+and the remaining leading order non-trivial leaves have the same decay by applying the partial expectation
+method (Subsection 5.13 in [17]).
+We conclude the proof.
+Proof of Lemma B.6. From the above version of an isotropic law, we also arrive at the isotropic version of
+the entrywise law in Lemma B.8 by taking A = Q + X and B = Q. Then, it is easy to check that
+wA = 2
+�
+1 + Eq
+�
+Vq
+�
+,
+wAB = 1 + Eq
+�
+Vq
+,
+wB = 1.
+Precisely, applying Lemma B.12 directly, we see that
+2⟨u, X(QT Q − zI)−1QT u⟩
+= ⟨u, X(QT Q − zI)−1QT u⟩ + ⟨u, Q(QT Q − zI)−1XT u⟩
+= ⟨u, A(BT B − zI)−1AT u⟩ − ⟨u, B(BT B − zI)−1BT u⟩ − ⟨u, X(BT B − zI)−1XT u⟩
+= 2s(z)
+�
+1 + Eq
+�
+Vq
+�
++ zs(z)s(z)2
+�
+1 + Eq
+�
+Vq
+�2
+− s(z) − zs(z)s(z)2 − s(z) − zs(z)s(z)2 E2
+q
+Vq
+= 2 Eq
+�
+Vq
+(s(z) + zs(z)s(z)2) = 2 Eq
+�
+Vq
+(zs(z) + 1)
+with O≺(N −φ) error terms, and it exactly matches the entrywise law since correlation wXQ =
+Eq
+√
+Vq . Thus,
+we conclude that the improved PCA via the entrywise transform holds for the spike U s.t. ∥U T U − Ik∥F ,
+∥U∥∞ ≺ N −φ, where φ > 1/4.
+C
+Proof of CLTs
+In Appendix C, we prove the CLT for the LSS of spiked random matrices. The proof of the CLT for the
+LSS is based on the strategy of [6] in which the LSS is first written as a contour integral of the resolvent of
+a spiked Wigner matrix. Then, the averaged trace of the resolvent converges to a Gaussian process, which
+also implies that the limiting distribution of the LSS is Gaussian.
+It is the biggest obstacle in adapting the proof in [6] for spiked matrices that the martingale CLT and
+covariance computation are hard to be reproduced with spikes; even with the special choice of rank-1 spike
+the proof for the CLT is very tedious as in [9]. In [22], the interpolation between a general rank-1 spike
+54
+
+and the special rank-1 spiked was introduced to compare the LSS, based on an ansatz that the mean and
+the variance of the LSS do not depend on the choice of the spike. In this paper, since we do not have a
+reference matrix to be compared with as in the rank-1 case, we introduce a direct interpolation between a
+spiked random matrices of general rank and a matrix without any spikes. With the interpolation, we find
+the change of the mean in the limiting Gaussian distribution and also prove that its variance is invariant.
+C.1
+Proof of CLTs for spiked random matrices
+Proof of Theorem 5.2. We adapt the proof of Theorem 5 in [22] with the following change. Instead of inter-
+polating the spiked Wigner matrices M with the original signal and with the signal with all 1’s considered in
+[9], we directly interpolate M and W and track the change of the mean. Consider the following interpolating
+matrix
+M(θ) = θ
+√
+λUU T + W
+and the corresponding eigenvalues {µi(θ)}N
+i=1 of M(θ) for θ ∈ [0, 1]. Let Γ be a rectangular contour in the
+proof of Theorem 5 in [22]. Applying Cauchy’s integral formula, we have
+N
+�
+i=1
+f(µi(1)) − N
+� 2
+−2
+√
+4 − x2
+2π
+f(x) dx = − N
+2πi
+�
+Γ
+f(z)
+�
+sN(1, z) − ssc(z)
+�
+dz
+(C.1)
+where ssc(z) = −z+
+√
+z2−4
+2
+is the Stieltjes transform of the Wigner semicircle law and sN(θ, z) is the Stieltjes
+transform of the empirical spectral distribution (ESD) of M(θ) for θ ∈ [0, 1]. Note that the normalized trace
+of the resolvent satisfies
+1
+N Tr R(θ, z) = 1
+N
+N
+�
+i=1
+1
+µi(θ) − z = sN(θ, z)
+(C.2)
+where R(θ, z) is the resolvent corresponding to M(θ), defined as
+R(θ, z) := (M(θ) − zI)−1
+(C.3)
+for z ∈ C+ and θ ∈ [0, 1].
+The change of the mean in the CLT for W and the CLT for M can be computed by tracking the change
+of the corresponding resolvent in (C.3), since (C.1) can be decomposed by
+N
+�
+i=1
+f(µi(1)) − N
+� 2
+−2
+√
+4 − x2
+2π
+f(x) dx = − 1
+2πi
+�
+Γ
+f(z)
+�
+Tr R(1, z) − Tr R(0, z)
+�
+dz
+(C.4)
+− 1
+2πi
+�
+Γ
+f(z)
+�
+Tr R(0, z) − Nssc(z)
+�
+dz
+(C.5)
+and the fluctuation result of (C.5) is already given in [6].
+Set Γε = {z ∈ C : minw∈Γ |z − w| ≤ ε}. Choose ε so that
+min
+w∈Γε,x∈[−2,2] |x − w| > 2ε.
+55
+
+Following the proof of Theorem 5 in [22], on z ∈ Γε
+1/2 := Γε ∩ {z ∈ C : |Imz| > N −1/2}, we first find that
+∂
+∂θ Tr R(θ, z) = −
+k
+�
+m=1
+√
+λ ∂
+∂z
+�
+x(m)T R(θ, z)u(m)
+�
+= −k ∂
+∂z
+�
+√
+λssc(z)
+1 + θ
+√
+λssc(z)
+�
++ O(N − 1
+2 )
+= −
+k
+√
+λs′
+sc(z)
+(1 + θ
+√
+λssc(z))2 + O(N − 1
+2 )
+(C.6)
+with high probability. More precisely, since the elementary resolvent expansion implies
+R(0, z) − R(θ, z) = θ
+√
+λR(θ, z)
+� k
+�
+ℓ=1
+u(ℓ)u(ℓ)T
+�
+R(0, z),
+(C.7)
+we then find that
+�
+u(m)T R(0, z)u(m)
+�
+=
+�
+u(m)T R(θ, z)u(m)
+�
++ θ
+√
+λ
+k
+�
+ℓ=1
+�
+u(m)T R(θ, z)u(ℓ)
+� �
+u(ℓ)T R(0, z)u(m)
+�
+.
+From the rigidity of the eigenvalues, we have a deterministic bound for resolvent
+|
+�
+u(m)T R(θ, z)u(ℓ)
+�
+| ≤ ∥R(θ, z)∥ ≤ C.
+(C.8)
+Since columns of spike {u(ℓ)}k
+ℓ=1 are orthonormal, the isotropic local law for R(0, z) implies that
+�
+u(m)T R(0, z)u(ℓ)
+�
+= s(z)δmℓ + O(N −1/2).
+(C.9)
+uniformly on z ∈ Γε. We then obtain that
+�
+u(m)T R(0, z)u(m)
+�
+=
+�
+u(m)T R(θ, z)u(m)
+� ���
+1 + θ
+√
+λ
+�
+u(m)T R(0, z)u(m)
+��
++ O(N − 1
+2 )
+and so
+�
+u(m)T R(θ, z)u(m)
+�
+=
+ssc(z)
+1 + θ
+√
+λssc(z)
++ O(N − 1
+2 ).
+This proves (C.6).
+Moreover, on Γε, we easily check that the exactly same argument holds for a finite rank perturbation of
+Wigner matrix (e.g. interlacing and rigidity properties). Thus, we conclude that (C.4) is
+k
+2πi
+�
+Γ
+√
+λs′
+sc(z)
+1 +
+√
+λssc(z)
+f(z)dz + o(1)
+with high probability.
+Finally, following the computation in the proof of Lemma 4.4 in [9], we then find that the difference
+between the LSS of M and the LSS of W is
+k
+∞
+�
+ℓ=1
+√
+λℓτℓ(f).
+(C.10)
+This proves the desired theorem.
+56
+
+Proof of Theorem 5.5. The proof of the CLT for the spiked rectangular matrices is quite similar to the case
+of spiked Wigner matrix. We first consider the interpolating matrix for the additive model, defined as
+Y (θ) = θ
+√
+λUV T + X
+(C.11)
+for θ ∈ [0, 1]. Note that Y (0) = X and Y (1) = Y . Denote by µ1(θ) ≥ µ2(θ) ≥ · · · ≥ µM(θ) the eigenvalues
+of Y (θ)Y (θ)T . We also define the resolvent
+G(θ, z) = (Y (θ)Y (θ)T − zI)−1,
+G(θ, z) = (Y (θ)T Y (θ) − zI)−1
+(C.12)
+for z ∈ C.
+We choose (N-independent) constants a− < d−, a+ > d+, and v0 ∈ (0, 1) so that the function f is
+analytic on the rectangular contour Γ whose vertices are (a− ± iv0) and (a+ ± iv0). With overwhelming
+probability, all eigenvalues of Y (θ)Y (θ)T are contained in Γ. Applying Cauchy’s integral formula, we find
+that
+M
+�
+i=1
+f(µi(1)) −
+M
+�
+i=1
+f(µi(0)) = −
+� 1
+2πi
+�
+Γ
+f(z) (Tr G(1, z) − Tr G(0, z)) dz
+�
+(C.13)
+To estimate the difference Tr G(1, z) − Tr G(0, z), we consider its derivative
+∂
+∂θ Tr G(θ, z). Note that
+∂Gab(θ)
+∂Yij(θ) = −Gai(θ)(Y (θ)T G(θ))jb − (G(θ)Y (θ))ajGib(θ),
+dYij(θ)
+dθ
+=
+√
+λuivT
+j .
+(C.14)
+Thus, by chain rule
+∂
+∂θ Tr G(θ, z) =
+M
+�
+a=1
+M
+�
+i=1
+N
+�
+j=1
+∂Yij(θ)
+∂θ
+∂Gaa(θ)
+∂Yij(θ)
+= −
+M
+�
+a=1
+M
+�
+i=1
+N
+�
+j=1
+√
+λuivT
+k [Gai(θ)(Y (θ)T G(θ))ja + (G(θ)Y (θ))ajGia(θ)]
+= −2
+M
+�
+a=1
+M
+�
+i=1
+N
+�
+j=1
+M
+�
+b=1
+√
+λuivT
+j [Ybj(θ)Gba(θ)Gai(θ)]
+(C.15)
+From the fact
+� ∂
+∂z G(θ)
+�
+bi
+= (G(θ)2)bi =
+�
+a
+Gba(θ)Gai(θ),
+we then find that
+∂
+∂θ Tr G(θ, z) = −2
+√
+λ ∂
+∂z
+M
+�
+i=1
+N
+�
+j=1
+uivT
+j (G(θ)Y (θ))ij = −2
+√
+λ ∂
+∂z
+k
+�
+ℓ=1
+⟨u(ℓ), G(θ)Y (θ)v(ℓ)⟩.
+(C.16)
+It remains to estimate
+∂
+∂z⟨u(ℓ), G(θ)Y (θ)v(ℓ)⟩ for 1 ≤ ℓ ≤ k. We suffices to estimate the desired term
+for fixed ℓ. From now, we omit ℓ-dependency. Note that
+⟨u, G(θ)Y (θ)v⟩ = θ
+√
+λ⟨u, G(θ)u⟩ + ⟨u, G(θ)Xv⟩.
+57
+
+We consider the resolvent expansion
+G(0, z) − G(θ, z) = G(θ, z) (H(θ) − H(0)) G(0, z)
+= G(θ, z) (θ2λuuT + θ
+√
+λXvuT + θ
+√
+λuvT XT ) G(0, z).
+(C.17)
+Taking inner products with u and v, we obtain
+⟨u, G(0)u⟩ = ⟨u, G(θ)u⟩ + θ2λ⟨u, G(θ)u⟩⟨u, G(0)u⟩
++ θ
+√
+λ⟨u, G(θ)Xv⟩⟨u, G(0)u⟩ + θ
+√
+λ⟨u, G(0)Xv⟩⟨u, G(θ)u⟩
+(C.18)
+and
+⟨u, G(0)Xv⟩ = ⟨u, G(θ)Xv⟩ + θ2λ⟨u, G(θ)Xv⟩⟨u, G(0)Xv⟩
++ θ
+√
+λ⟨u, G(θ)Xv⟩⟨u, G(0)Xv⟩ + θ
+√
+λ⟨v, XT G(0)Xv⟩⟨u, G(θ)u⟩,
+(C.19)
+where we omitted z-dependence for brevity. We then use the following result to control the terms in (C.18)
+and (C.19). Recall the definition of s(z) and s(z) in Lemmas B.4 and B.11. Moreover, we consider the same
+linearization HX(z) of the matrix X and its inverse RX(z) = HX(z)−1 as in (B.33) and (B.34).
+Lemma C.1 (Isotropic local law). For an N-independent constant ε > 0, let Γε be the ε-neighborhood of Γ,
+i.e.,
+Γε = {z ∈ C : min
+w∈Γ |z − w| ≤ ε}.
+Choose ε small so that the distance between Γε and [d−, d+] is larger than 2ε, i.e.,
+min
+w∈Γε,x∈[d−,d+] |x − w| > 2ε.
+(C.20)
+Then, for any unit vectors x, y ∈ CM+N independent of X,
+|⟨x, (RX(z) − Π(z))y⟩| ≺ N −1/2,
+(C.21)
+uniformly on z ∈ Γε, where
+Π(z) =
+�s(z) · IM
+0
+0
+zs(z) · IN
+�
+.
+(C.22)
+Proof. See Theorems 3.6, 3.7, Corollary 3.9, and Remark 3.10 in [37]. Note that Im s(z), Im s(z) = Θ(η) on
+the vertical part of Γε, i.e., the neighborhood of the line segment joining (a++iv0) and (a+−iv0) (respectively
+(a− + iv0) and (a− − iv0)).
+Set
+A := ⟨u, G(0, z)u⟩,
+B := ⟨u, G(0, z)Xv⟩,
+C := ⟨v, XT G(0, z)Xv⟩.
+Recall that
+RX(z) =
+� G(0, z)
+G(0, z)X
+XT G(0, z)
+zG(0, z)
+�
+.
+(C.23)
+Then, as consequences of Lemma C.1 with appropriate choices of the deterministic vectors,
+A = s(z) + O≺(N −1/2),
+C = ⟨v, zG(0, z)v⟩ + 1 + O(N −1/2) = d0(zs(z) + 1) + O≺(N −1/2),
+(C.24)
+58
+
+and
+B = O≺(N −1/2).
+We thus have from (C.18) and (C.19) that
+⟨u, G(θ)Xv⟩ = −θd0
+√
+λs(z)(zs(z) + 1)
+θ2λzs(z) + θ2λ + 1
++ O≺(N −1/2)
+⟨u, G(θ)u⟩ =
+s(z)
+θ2λzs(z) + θ2λ + 1 + O≺(N ���1/2)
+(C.25)
+and hence
+⟨u, G(θ)Y (θ)v⟩ = θ
+√
+λ⟨u, G(θ)u⟩ + ⟨u, G(θ)Xv⟩ =
+θ
+√
+λzs(z) + θ
+√
+λ
+θ2λzs(z) + θ2λ + 1 + O≺(N −1/2).
+(C.26)
+Note that this estimate is uniform on θ. Differentiating it with respect to z and plugging it back to (C.16),
+we get
+∂
+∂θ Tr G(θ, z) = −k
+2θλ d
+dz(zs(z) + 1)
+(θ2λzs(z) + θ2λ + 1)2 + O≺(N −1/2)
+and, integrating over θ, we obtain
+Tr G(1, z) − Tr G(0, z) =
+� 1
+0
+∂
+∂θ Tr G(θ, z)dθ = −k
+d
+dzλ(zs(z) + 1)
+λzs(z) + λ + 1 + O≺(N −1/2).
+(C.27)
+We now invoke the following relation between the Stieltjes transforms for Marchenko–Pastur law and the
+Wigner semicircle law. Let
+ssc(z) = −z +
+√
+z2 − 4
+2
+be the Stieltjes transform of the Wigner semicircle law and
+ϕ(z) =
+1
+√d0
+(z − (1 + d0)).
+Then
+�
+d0(zs(z) + 1) = ssc(ϕ(z)).
+(C.28)
+We thus have
+1
+2πi
+�
+Γ
+f(z)λ d
+dz(zs(z) + 1)
+λzs(z) + λ + 1 dz =
+1
+2πi
+�
+Γ
+�f(ϕ(z)) λs′
+sc(ϕ(z))ϕ′(z)
+λssc(ϕ(z)) + √d0
+dz
+=
+1
+2πi
+�
+�Γ
+�f(ϕ)
+λs′
+sc(ϕ)
+λssc(ϕ) + √d0
+dϕ
+(C.29)
+where we let f(√d0z + 1 + d0) = �f(z) and �Γ = ϕ(Γ). (Note that �Γ contains the interval [−2, 2].)
+So far, we have proved that
+M
+�
+i=1
+f(µi(1)) −
+M
+�
+i=1
+f(µi(0)) =
+k
+2πi
+�
+�Γ
+�f(ϕ)
+λs′
+sc(ϕ)
+λssc(ϕ) + √d0
+dϕ + O≺(N −1/2).
+(C.30)
+Since the difference in (C.30) is the sum of a deterministic term and a random term stochastically dominated
+59
+
+by N −1/2, we can see that the CLT holds for the LSS with the non-null model Y (1). Moreover, the variance
+is the same as that of the null model, which is
+VY (f) = 2
+∞
+�
+ℓ=1
+ℓτℓ( �f)2 + (w4 − 3)τ1( �f)2.
+(C.31)
+(See, e.g., [10].)
+The change of the mean is the first term in the right side of (C.30), which can be computed by following
+the proof of Lemma 4.4 in [9]. We obtain
+mY (f) =
+�f(2) + �f(−2)
+4
+− 1
+2τ0( �f) + (w4 − 3)τ2( �f) + k
+∞
+�
+ℓ=1
+� λ
+√d0
+�ℓ
+τℓ( �f).
+(C.32)
+This proves the first part of Theorem 5.2 for the additive model.
+For the multiplicative model, we will follow the same strategy as in the additive model. Let
+Y (θ) = X + θγUU T X
+(C.33)
+for θ ∈ [0, 1].
+Note that Y (0) = X and Y (1) = Y .
+We denote by µ1(θ) ≥ µ2(θ) ≥ · · · ≥ µM(θ) the
+eigenvalues of Y (θ)Y (θ)T , and also let
+G(θ, z) = (Y (θ)Y (θ)T − zI)−1,
+G(θ, z) = (Y (θ)T Y (θ) − zI)−1
+(C.34)
+for z ∈ C. We have the relations
+∂Gab(θ)
+∂Yij(θ) = −Gai(θ)(Y (θ)T G(θ))jb − (G(θ)Y (θ))ajGib(θ),
+∂Yij(θ)
+∂θ
+= γ
+M
+�
+c=1
+uiuT
+b Xbj.
+(C.35)
+Following (C.15)-(C.16), we get
+∂
+∂θ Tr G(θ, z) = −γ
+M
+�
+a=1
+M
+�
+i=1
+N
+�
+j=1
+M
+�
+b=1
+uiuT
+b Xbj[Gai(θ)(Y (θ)T G(θ))ja + (G(θ)Y (θ))ajGia(θ)]
+= −2γ
+M
+�
+a=1
+M
+�
+i=1
+N
+�
+j=1
+M
+�
+b=1
+uiuT
+b Xbj[(Y (θ)T G(θ))jaGai(θ)]
+= −2γ ∂
+∂z
+M
+�
+i=1
+N
+�
+j=1
+M
+�
+b=1
+uiuT
+b Xbj(G(θ)Y (θ))ij
+= −2γ ∂
+∂z
+k
+�
+ℓ=1
+⟨u(ℓ), G(θ)Y (θ)XT u(ℓ)⟩ = −2γ ∂
+∂z
+k
+�
+ℓ=1
+⟨u(ℓ), G(θ)Y (θ)Y (0)T u(ℓ)⟩.
+(C.36)
+Moreover, since
+Y (0) = X = (I + θγUU T )−1Y (θ) =
+�
+I −
+θγ
+1 + θγ UU T
+�
+Y (θ),
+(C.37)
+60
+
+we have
+⟨u(ℓ), G(θ)Y (θ)Y (0)T u(ℓ)⟩ = ⟨u(ℓ), G(θ)Y (θ)Y (θ)T (I + θγUU T )−1u(ℓ)⟩
+= ⟨u(ℓ), (I + zG(θ))(I + θγUU T )−1u(ℓ)⟩
+=
+1
+1 + θγ +
+z
+1 + θγ ⟨u(ℓ), G(θ)u(ℓ)⟩.
+(C.38)
+To estimate the term ⟨u(ℓ), G(θ)u(ℓ)⟩, we use the following Anisotropic local law in [37].
+Lemma C.2 (Anisotropic local law). Let Γε be the ε-neighborhood of Γ as in Lemma C.1. Then, for any
+unit vectors x, y ∈ CM independent of X, the following estimate holds uniformly on z ∈ Γε :
+����
+�
+x,
+�
+G(θ, z) +
+�
+zI + zs(z)(I + θγUU T )2�−1�
+y
+����� ≺ N − 1
+2 .
+(C.39)
+Proof. The proof of Lemma C.2 is the same as that of Lemma C.1.
+Now, as in the additive case, we drop the ℓ-dependency. From Lemma C.2, we find that
+⟨u, G(θ)u⟩ = −
+�
+u,
+�
+zI + zs(z)(I + θγUU T )2�−1
+u
+�
++ O(N −1/2)
+= −
+1
+(1 + θγ)2z(1 + s(z)) + O(N −1/2),
+(C.40)
+and plugging it into (C.38), we obtain
+⟨u, G(θ)Y (θ)Y (0)T u⟩ =
+1
+1 + θγ −
+1
+(1 + θγ)(1 + (1 + θγ)2s(z)) + O(N −1/2).
+(C.41)
+We thus get
+∂
+∂θ Tr G(θ, z) = −2kγ
+(1 + θγ)s′(z)
+(1 + (1 + θγ)2s(z))2 + O(N −1/2),
+(C.42)
+and integrating it yields
+Tr G(1, z) − Tr G(0, z) = −k
+λs′(z)
+(1 + s(z))(1 + (1 + λ)s(z)) + O(N −1/2)
+= −λk d
+dz(zs(z) + 1)
+λzs(z) + λ + 1 + O(N −1/2).
+(C.43)
+Since (C.43) coincides with (C.27), the rest of the proof is exactly the same as in the additive case. This
+finishes the proof of the first part of Theorem 5.2.
+C.2
+Proof of CLTs for entrywise transformed matrices
+Proof of Theorem 5.3. We adapt the proof of Theorem 7 in [22] with the following changes. Let S be the
+variance matrix of the transformed matrix �
+M. We then find that
+Sij = E[�
+M 2
+ij] − (E[�
+Mij])2 = 1
+N + λ(GH − Fg)(uiuT
+j )2 + O(N 1−8φ)
+and
+Sii = E[�
+M 2
+ii] − (E[�
+Mii])2 = w2
+N + λ(Gg,d − Fg,d)(uiuT
+i )2 + O(N 1−8φ).
+61
+
+Normalizing and centering each entry of the matrix �
+M, we arrive at another Wigner matrix �
+W where
+�
+Wij =
+1
+�
+NSij
+(�
+Mij − E�
+Mij),
+�
+Wii =
+� w2
+NSii
+(�
+Mii − E�
+Mii).
+Interpolating �
+W and �
+M − E[�
+M] by �
+W(θ) = (1 − θ)�
+W + θ(�
+M − E[�
+M]), �
+W(θ) is a general Wigner-type matrix
+with the corresponding quadratic vector equation
+−
+1
+mi(θ, z) = z +
+N
+�
+j=1
+E[�
+Wij(θ)2] · mj(θ, z)
+where mi(θ, z)δij is the limiting distribution of the (i, j)-element of the resolvent
+R
+�
+W (θ, z) = (�
+W(θ) − zI)−1
+for 0 ≤ θ ≤ 1. Recall the ssc(z) is the Stieltjes transform of the Wigner semicircle law. We also directly
+check that mi(θ, z) = ssc(z) + C1(uiuT
+i ) + C2N −1 = ssc(z) + O(N −2φ). Moreover, the anisotropic local law
+for the general Wigner-type matrix in [2] implies that uniformly on z ∈ Γε
+1/2
+(u(m)T R
+�
+W (θ, z)u(ℓ)) = ssc(z)δmℓ + O(N −1/2).
+Following the proof of Lemmas B.2 and B.3 in [22], we check that
+• Uniformly on z ∈ Γε
+1/2,
+Tr R
+�
+W (1, z) − Tr R
+�
+W (0, z) = kλ(GH − Fg)s′
+sc(z)ssc(z) + O(N 1N −4φ)
+(C.44)
+• Uniformly on z ∈ Γε\Γε
+1/2,
+| Tr R
+�
+W (1, z) − Tr R
+�
+W (0, z)| = O(N 1N −2/3).
+(C.45)
+Compared with the bound shown in [22], we give the following remark:
+• The error bound in (C.44) is better. This sharper bound can be obtained by using the fact �
+a ua(ℓ)2 =
+1 instead of
+���
+a ua(ℓ)2�� ≤ N∥u(ℓ)∥2
+∞.
+Our next step is to consider �
+M = �
+W(1) + E[�
+M]. Since
+�
+M = �
+W(1) +
+�
+λFgUU T + diag(d1, · · · , dN) + E
+where di = E[�
+Mii] −
+�
+λFg(UU T )ii, we then find that
+Tr(�
+M − zI)−1 − Tr R
+�
+W (0, z)
+= kλ(GH − Fg)s′
+sc(z)ssc(z) −
+k
+�
+λFgs′
+sc(z)
+1 +
+�
+λFgssc(z) − k
+√
+λ(
+�
+Fg,d −
+�
+Fg)s′
+sc(z) + O(N −1/2)
+uniformly on z ∈ Γε
+1/2. Thus, we obtain the desired CLT by applying Cauchy’s integral formula as in the
+proof of Theorem 5.2.
+62
+
+Proof of Theorem 5.6. Since the proof of the transformed CLT for the spiked Wigner matrix follows the
+proof in [22], we only describe the process briefly. On the other hand, there is no technical reference for the
+spiked rectangular matrices. As we mentioned before, our consideration is only the additive case.
+We consider the optimal entrywise transformation defined by a function
+h(w) := −g′(w)
+g(w) .
+(C.46)
+If λ = 0, it is immediate to see that for all i, j
+E[h(
+√
+NYij)] =
+� ∞
+−∞
+h(w)g(w)dw = −
+� ∞
+−∞
+g′(w)dw = 0.
+Further, with λ = 0, as shown in Proposition 4.2 of [50],
+Fg := E[h(
+√
+NYij)2] =
+� ∞
+−∞
+h(w)2g(w)dw =
+� ∞
+−∞
+g′(w)2
+g(w) dw ≥ 1,
+(C.47)
+where the equality holds if and only if
+√
+NXij is a standard Gaussian (hence h(w) = w).
+We define a transformed matrix �Y as follows: the terms of �Y are defined by
+�Yij =
+1
+�
+FgN h(
+√
+NYij).
+(C.48)
+Note that the entries of �Y are independent up to symmetry. Since g is smooth, h is also smooth and all
+moments of
+√
+N �Yij are O(1). Thus, applying a high-order Markov inequality, it is immediate to find that
+�Yij = O(N − 1
+2 ).
+C.2.1
+Decomposition of the transformed matrix
+We first estimate the mean and the variance of entry by using the comparison method with the pre-
+transformed entries. For all i, j, we find that
+E[�Yij] =
+1
+�
+FgN
+� ∞
+−∞
+h(w)g
+�
+w −
+√
+NλuivT
+j
+�
+dw
+= −
+1
+�
+FgN
+� ∞
+−∞
+g′(w)
+g(w)
+�
+g
+�
+w −
+√
+NλuivT
+j
+�
+− g(w)
+�
+dw.
+(C.49)
+In the Taylor expansion
+g
+�
+w −
+√
+NλuivT
+j
+�
+− g(w)
+=
+4
+�
+ℓ=1
+g(ℓ)(w)
+ℓ!
+�
+−
+√
+NλuivT
+j
+�ℓ
++
+g(5) �
+w − θ
+√
+NλuivT
+j
+�
+5!
+�
+−
+√
+NλuivT
+j
+�5
+(C.50)
+63
+
+for some θ ∈ (0, 1). Note that the second term and the fourth term in the summation are even functions.
+Since g′/g is an odd function, we find that
+E[�Yij] =
+1
+�
+Fg
+√
+λuivT
+j
+� ∞
+−∞
+g′(w)2
+g(w) dw + C3N
+�√
+λuivT
+j
+�3
++ O(N 2(uivT
+j )5)
+=
+�
+λFguivT
+j + C3N
+�√
+λuivT
+j
+�3
++ O(N 2(uivT
+j )5)
+(C.51)
+for some (N-independent) constant C3. Similarly, since
+�
+g′
+g
+�2
+is even,
+E[�Y 2
+ij] =
+1
+FgN
+� ∞
+−∞
+�g′(w)
+g(w)
+�2
+g
+�
+w −
+√
+NλuivT
+j
+�
+dw
+= 1
+N +
+1
+FgN
+� ∞
+−∞
+�g′(w)
+g(w)
+�2 �
+g
+�
+w −
+√
+NλuivT
+j
+�
+− g(w)
+�
+dw
+= 1
+N +
+1
+2Fg
+�√
+λuivT
+j
+�2 � ∞
+−∞
+g′(w)2g′′(w)
+g(w)2
+dw + O(N(uivT
+j )4)
+= 1
+N + λGH(uivT
+j )2 + O(N(uivT
+j )4).
+(C.52)
+where
+GH =
+1
+2Fg
+� ∞
+−∞
+g′(w)2g′′(w)
+g(w)2
+dw.
+The evaluation of the mean and the variance shows that the transformed matrix �Y is not a spiked
+rectangular matrix when λ > 0, since the variances of the entries are not identical.
+Our strategy is to
+approximate �Y as a spiked generalized rectangular Gram matrix for which the variances of the each entries
+is 1/N in high-dimensional regime. Let S be the variance matrix of �Y defined as
+Sij = E[�Y 2
+ij] − (E[�Yij])2.
+(C.53)
+From (C.51) and (C.52),
+Sij = 1
+N + (GH − Fg)
+�√
+λuivT
+j
+�2
++ O(N∥U∥4
+∞∥V ∥4
+∞),
+(C.54)
+which shows that �Y is indeed approximately a spiked generalized Gram matrix.
+C.2.2
+CLT for a random Gram matrix
+We use the local law for general rectangular Gram matrices in [4]. Consider an another M × N rectangular
+matrix A = (Aij) defined by
+Aij =
+1
+�
+NSij
+(�Yij − E[�Yij]).
+(C.55)
+Note that E[Aij] = 0, E[A2
+ij] = 1
+N . Then the matrix A is a usual rectangular matrix. We set
+GA(z) = (AAT − zI)−1
+(z ∈ C+).
+(C.56)
+64
+
+Next, we introduce an interpolation for A. For 0 ≤ θ ≤ 1, we define a matrix A(θ) by
+Aij(θ) = (1 − θ)Aij + θ(�Yij − E[�Yij]) =
+�
+1 − θ + θ
+�
+NSij
+�
+Aij
+=
+�
+1 + θNλ(GH − Fg)(uivT
+j )2
+2
++ O(N 2(uivT
+j )4)
+�
+Aij
+(C.57)
+Note that A(0) = A and A(1) = �Y − E[�Y ]. For 0 ≤ θ ≤ 1, A(θ) is a random Gram matrix considered in
+[4] satisfying the conditions (A)–(D) therein. Moreover, if we let
+GA(θ, z) = (A(θ)A(θ)T − zI)−1
+(z ∈ C+)
+(C.58)
+and Sij(θ) = E[Aij(θ)2], then Theorem 1.7 of [4] asserts that the limiting distribution of GA
+ij(z) is si(z)δij,
+where si(θ, z) is the unique solution to the system of quadratic vector equations
+−
+1
+si(θ, z) = z +
+N
+�
+j=1
+Sij(θ) zsj(θ, z)
+(C.59)
+and
+−
+1
+sj(θ, z) = z +
+M
+�
+i=1
+Sij(θ) zsi(θ, z)
+(C.60)
+Remark C.3. Recall that s(z) is the Stieltjes transform of the Marchenko-Pastur measure. We can then
+find that si(θ, z) = s(z)+C1(uiuT
+i )+C2N −1 = s(z)+O(N −1/2) and sj(θ, z) = s(z)+C1(vjvT
+j )+C2N −1 =
+s(z) + O(N −1/2); see also Lemma 3.9 of [4].
+For the resolvent GA(θ, z), we will use the following lemma for the random Gram matrix:
+Lemma C.4 (Anisotropic local law for random Gram matrix). Let Γε be the ε-neighborhood of Γ as in
+Lemma C.1. Then, for any deterministic x = (x1, . . . , xM), y = (y1, . . . , yM) ∈ CM with ∥x∥ = ∥y∥ = 1,
+the following estimate holds uniformly on z ∈ Γε ∩ {z ∈ C+ : Im z > N − 1
+2 }:
+������
+M
+�
+i=1
+M
+�
+j=1
+xiGA
+ij(θ, z)yj −
+M
+�
+i=1
+si(θ, z)xiyi
+������
+= O(N − 1
+2 ).
+(C.61)
+and, for any deterministic x = (x1, . . . , xN), y = (y1, . . . , yN) ∈ CN with ∥x∥ = ∥y∥ = 1,
+������
+N
+�
+i=1
+N
+�
+j=1
+xiGA
+ij(θ, z)yj −
+N
+�
+i=1
+si(θ, z)xiyi
+������
+= O(N − 1
+2 ).
+(C.62)
+Proof of Lemma C.4. Let Ψ(z) =
+�
+1
+M Im z be the control parameter for the random gram matrix model. We
+then note that the bound for the entrywise local law is N −1/2 since Ψ(z) ≺ N −1/2 on Γε ∩ {z ∈ C+ : Im z >
+N − 1
+2 }. With the entrywise local law in [4], the proof of the anisotropic law exactly follows the maximal
+65
+
+expansion argument used in [2, 17] and Lemma B.12. We consider the following decomposition of (C.61):
+M
+�
+i̸=j
+xiGA
+ij(θ, z)xj +
+M
+�
+i=1
+(GA
+ii − si(θ, z))xixi.
+(C.63)
+From now, we drop A, θ and z-dependencies for brevity and use the linearization matrix HA(θ)(z) ≡ H and
+its inverse R. Then, in usual, we suffices to prove that
+Z ≡
+�
+a̸=b
+xaRabxb ≺ N −1/2.
+To prove the above high probability bound, we will bound the 2p-moments E[|Z|2p] ≺ N −p/2 by deriving
+the maximally expanded form via the resolvent identity in Lemma B.10.
+Now, we will check the representation of the maximally expanded diagonal resolvent elements. e.g. R(B\b)
+bb
+,
+b ∈ B. Together with Remark C.3, we then conclude that the standard argument in [2] is valid for our model.
+By applying Shur’s complement lemma and (C.59), for b ∈ B,
+1
+R(B\b)
+bb
+= −z −
+(B)
+�
+α,β
+HbαR(B)
+αβ Hβb
+=
+1
+sb(θ, z) +
+�
+β
+Sbβ (zsβ(θ, z)) −
+(B)
+�
+α,β
+HbαR(B)
+αβ Hβb
+=
+1
+sb(θ, z) −
+(B)
+�
+β
+(HbβR(B)
+ββ Hβb − Sbβ (zsβ(θ, z))) −
+(B)
+�
+α̸=β
+HbαR(B)
+αβ Hβb.
+(C.64)
+Then (C.64) and the analogue representation of R(B\β)
+ββ
+for β ∈ B replace the (6.2) in [2].
+With linearization H and its inverse R, one useful by-product of the above argument is
+⟨x, GA(θ)A(θ)y⟩ =
+�
+a
+�
+α
+xaRaαyα ≺ N −1/2.
+(C.65)
+Note that our model satisfies the closeness condition (A3) of Assumption 2.2 in [3] (See also Remark 2.4
+therein). On Γ\Γε
+1/2, we use the following results on the rigidity of eigenvalues.
+Lemma C.5 (Rigidity of eigenvalues for the random Gram matrix). Denote by µA
+1 (θ) ≥ µA
+2 (θ) ≥ · · · ≥
+µA
+M(θ) the eigenvalues of A(θ)A(θ)T . Let γi be the classical location of the eigenvalues with respect to the
+Marchenko-Pastur measure defined by
+� ∞
+γi
+ρMP,d0(dx) = 1
+M
+�
+i − 1
+2
+�
+(C.66)
+for i = 1, 2, . . . , M. Then,
+|µA
+i (θ) − γi| = O(M −2/3).
+(C.67)
+Proof. Note that the rigidity of the eigenvalues with an error of at most O(M −2/3) holds for random gram
+matrices at the classical location of the eigenvalues with respect to the probability measure ρ from the Stieltjes
+transform sρ(z) :=
+1
+M
+�
+i si(z), see Lemma 4 in [26]. Moreover, since |si(θ, z) − s(z)|, |sj(θ, z) − s(z)| =
+66
+
+O(M −2φ) for all i and j, we also have the desired rigidity near the classical location of Marchenko-Pastur
+law ρMP,d0.
+Remark C.6. In fact, rigorous proofs of the rigidity and anisotropic law are not given in [4, 3]. However,
+as in the proof of anisotropic local law for general Wigner-type matrix in [2], the above lemmas may be proved
+by using the local laws in [4] and standard methods in [2] (Remark 2.10 in [4] and Remark 2.7 in [3].)
+On Γε
+1/2, as a simple corollary to Lemma C.4, we obtain
+��⟨x, GA(θ, z)y⟩ − s(z)⟨x, y⟩
+�� = O(N − 1
+2 ),
+(C.68)
+and
+��⟨x, GA(θ, z)y⟩ − s(z)⟨x, y⟩
+�� = O(N − 1
+2 ).
+(C.69)
+We have the following lemma for the difference between Tr GA(0, z) and Tr GA(1, z) on Γε
+1/2.
+Lemma C.7. Let GA(θ, z) be defined as in Equations (C.57) and (C.58). Then, the following holds uniformly
+for z ∈ Γε
+1/2:
+Tr GA(1, z) − Tr GA(0, z) = −λ(GH − Fg)k ∂
+∂z (zs(z) + 1) + O(N∥U∥2
+∞∥V ∥2
+∞).
+(C.70)
+We will prove Lemma C.7 later.
+From Lemma C.5, we find that
+| Tr GA(1, z) − Tr GA(0, z)| =
+�����
+N
+�
+i=1
+�
+1
+µA
+i (1) − z −
+1
+µA
+i (0) − z
+������ =
+�����
+N
+�
+i=1
+µA
+i (0) − µA
+i (1)
+(µA
+i (1) − z)(µA
+i (0) − z)
+�����
+≤
+�����
+N
+�
+i=1
+|µA
+i (0) − γi| + |γi − µA
+i (1)|
+(µA
+i (1) − z)(µA
+i (0) − z)
+����� = O(N 1/3)
+(C.71)
+uniformly for z ∈ Γ. Thus, from (C.70) and (C.71),
+1
+2πi
+�
+Γ
+f(z) Tr GA(1, z)dz − 1
+2πi
+�
+Γ
+f(z) Tr GA(0, z)dz
+=
+1
+2πi
+�
+Γε
+1/2
+f(z)
+�
+Tr GA(1, z) − Tr GA(0, z)
+�
+dz + 1
+2πi
+�
+Γ\Γε
+1/2
+f(z)
+�
+Tr GA(1, z) − Tr GA(0, z)
+�
+dz
+= −λ(GH − Fg)k
+2πi
+�
+Γε
+1/2
+f(z) ∂
+∂z (zs(z) + 1)dz + O(N∥U∥2
+∞∥V ∥2
+∞) + O(N −1/6)
+= −λ(GH − Fg)k
+2πi
+�
+Γ
+f(z) ∂
+∂z (zs(z) + 1)dz + o(1).
+(C.72)
+Furthermore, using the relation (C.28), we have
+1
+2πi
+�
+Γ
+f(z) ∂
+∂z (zs(z) + 1)dz =
+1
+2πi
+�
+Γ
+f(z) 1
+√d0
+s′
+sc(ϕ(z))ϕ′(z)dz
+=
+1
+2√d0πi
+�
+�Γ
+�f(ϕ)s′
+sc(ϕ)dϕ =
+1
+√d0
+τ1( �f).
+67
+
+C.2.3
+CLT for a random Gram matrix with a spike and small perturbation
+Recall that A(1) = �Y − E[�Y ]. Our next step in the approximation is to consider �Y = A(1) + E[�Y ]. Since
+E[�Y ] is not a matrix of rank k, we instead consider
+B(θ) = A(1) + θ
+�
+λFgUV T ,
+GB(θ, z) = (B(θ)B(θ)T − zI)−1
+(C.73)
+To prove this part of CLT, we will adapt the strategy for the proof of Theorem 5.5 with Lemmas C.4 and
+C.5. We then find that, uniformly for z ∈ Γε
+1/2,
+Tr GB(1, z) − Tr GB(0, z) = −k
+d
+dzλFg(zs(z) + 1)
+λFgzs(z) + λFg + 1 + O≺(N −φ),
+(C.74)
+since ∥U T U − Ik∥F , ∥V T V − Ik∥F ≺ N −φ. Using the rigidity (Lemma C.5) and the eigenvalue interlacing
+property, we have
+Tr GB′(1, z) − Tr GB′(0, z) = O(1)
+on Γ\Γ1/2
+and so
+1
+2πi
+�
+Γ
+f(z) Tr GB(1, z)dz − 1
+2πi
+�
+Γ
+f(z) Tr GB(0, z)dz
+= − k
+2πi
+�
+Γ
+f(z) λFg d
+dz(zs + 1)
+λFg(zs + 1) + 1dz + o(1).
+(C.75)
+The remaining part is to control an effect of small perturbation (E[�Y ] −
+�
+λFgUV T )ij = CN(uivT
+j )3 +
+O(N 2(uivT
+j )5). First, we let
+B′ = B(1) + CN(uivT
+j )3,
+GB′(z) = (B′(B′)T − zI)−1
+(C.76)
+For 1 ≤ ℓ1, ℓ2, ℓ3 ≤ k, we consider vectors u3 and v3 such that
+(u3(ℓ1, ℓ2, ℓ3))i = u3
+i (ℓ1, ℓ2, ℓ3) :=
+√
+Nui(ℓ1)ui(ℓ2)ui(ℓ3)
+and
+(v3(ℓ1, ℓ2, ℓ3))j = v3
+j (ℓ1, ℓ2, ℓ3) :=
+√
+Nvj(ℓ1)vj(ℓ2)vj(ℓ3).
+We then observe that B′ contains k3 additional small spikes:
+B′ = B(1) + C
+�
+ℓ1,ℓ2,ℓ3
+u3(ℓ1, ℓ2, ℓ3)v3(ℓ1, ℓ2, ℓ3)T
+where ∥u3(ℓ1, ℓ2, ℓ3)∥∞, ∥v3(ℓ1, ℓ2, ℓ3)∥∞ ≺ N 1/2−3φ.
+In the above point of view, we are able to consider B′ as another spiked Gram matrix model with two
+types of spikes u(ℓ)v(ℓ)T and u3(ℓ1, ℓ2, ℓ3)(v3(ℓ1, ℓ2, ℓ3))T . As before, for 0 ≤ θ ≤ 1, let
+B′(θ) = A(1) + θ
+�
+λFg
+�
+ℓ
+u(ℓ)v(ℓ)T + θC
+�
+ℓ1,ℓ2,ℓ3
+u3(ℓ1, ℓ2, ℓ3)(v3(ℓ1, ℓ2, ℓ3))T
+and
+GB′(θ, z) = (B′(θ)B′(θ)T − zI)−1.
+68
+
+Following the proof of Theorem 5.5, we have
+∂
+∂θ Tr GB′(θ, z) = −2
+�
+λFg
+∂
+∂z
+�
+ℓ
+⟨u(ℓ), GB′(θ, z)B′(θ)v(ℓ)⟩
+− 2C ∂
+∂z
+�
+ℓ1,ℓ2,ℓ3
+⟨u3(ℓ1, ℓ2, ℓ3), GB′(θ, z)B′(θ)v3(ℓ1, ℓ2, ℓ3)⟩,
+and it can be observed that the first term of the right-hand side is the leading order term, since ∥u3(ℓ1, ℓ2, ℓ3)∥∞, ∥v3(ℓ1, ℓ2, ℓ3)∥∞ ≺
+N 1/2−3φ < N −φ. Moreover, from the definition of B′(θ), the leading order term of ⟨u(ℓ), GB′(θ, z)B′(θ)v(ℓ)⟩
+is ⟨u(ℓ), GB′(θ, z)B′(0)v(ℓ)⟩+θ
+�
+λFg⟨u(ℓ), GB′(θ, z)u(ℓ)⟩, since ⟨v(ℓ1), v(ℓ2)⟩ = δℓ1ℓ2+O(N −φ), ⟨v(ℓ), v3(ℓ1, ℓ2, ℓ3)⟩ =
+O(N 1/2−2φ) and ⟨v3(ℓ1, ℓ2, ℓ3), v3(ℓ4, ℓ5, ℓ6)⟩ = O(N 1−4φ). Carrying out the remaining procedures presented
+in the proof of Theorem 5.5 and collecting the leading order terms, we eventually obtain
+Tr GB′(1, z) − Tr GB′(0, z) = −k
+d
+dzλFg(zs(z) + 1)
+λFgzs(z) + λFg + 1 + O≺(N 1/2−2φ)
+(C.77)
+uniformly for z ∈ Γε
+1/2. Here, we last apply Lemma C.4 and (C.65) for B′(0) = A(1). Further, on Γ\Γ1/2,
+from the rigidity and the interlacing property of the eigenvalues,
+Tr GB′(1, z) − Tr GB′(0, z) = O(1).
+(C.78)
+Thus, we conclude that
+1
+2πi
+�
+Γ
+f(z) Tr GB′(z)dz − 1
+2πi
+�
+Γ
+f(z) Tr GB′(0, z)dz
+= − k
+2πi
+�
+Γ
+f(z) λFg d
+dz(zs + 1)
+λFg(zs + 1) + 1dz + o(1).
+Furthermore, we set Eij = (�Y − B′)ij = O(N 2(uivT
+j )5). Then
+∥E∥ ≤ ∥E∥F = O(N 2∥U∥4
+∞∥V ∥4
+∞) = o(N −1)
+(C.79)
+for some φ > 3/8. This implies that
+1
+2πi
+�
+Γ
+f(z) Tr G
+�Y (z)dz − 1
+2πi
+�
+Γ
+f(z) Tr GB′(1, z)dz = o(1).
+Remark C.8. Under the assumption that φ > 3/8, we suffices to consider E[�Yij] up to O(N 2(uivT
+j )5) error.
+However, (C.77) and (C.78) are valid for any finite approximation of E[�Y ] as presented in (C.51), even for
+any φ > 1/4. This means that the condition φ > 3/8 can be improved by considering a higher order expansion
+of E[�Y ]. For example, if we consider
+E[�Y ] =
+�
+λFguivT
+j + C1N(uivT
+j )3 + C2N 2(uivT
+j )5 + O(N 3(uivT
+j )7),
+then it can be checked that the contributions of the second and third terms are negligible, and the error
+Eij = O(N 3(uivT
+j )7) is also negligible if φ > 1/3, since
+∥E∥ ≤ ∥E∥F = O(N 3∥U∥6
+∞∥V ∥6
+∞) ≺ N 3−12φ = o(N −1).
+69
+
+C.2.4
+Conclusion for the proof of pre-transformed CLT
+We are now ready to prove pre-transformed CLT. Denote by �µ1 ≥ �µ2 ≥ · · · ≥ �µN the eigenvalues of �Y �Y T .
+Recall that we denoted by µA
+1 (0) ≥ µA
+2 (0) ≥ · · · ≥ µA
+N(0) the eigenvalues of A(0)A(0)T . From Cauchy’s
+integral formula, we have
+M
+�
+i=1
+f(�µi) − M
+� d+
+d−
+f(x) ρMP,d0(dx)
+=
+� M
+�
+i=1
+f(µA
+i (0)) −
+� d+
+d−
+f(x) ρMP,d0(dx)
+�
++
+� M
+�
+i=1
+f(�µi) −
+M
+�
+i=1
+f(µA
+i (0))
+�
+=
+� M
+�
+i=1
+f(µA
+i (0)) − M
+� d+
+d−
+f(x) ρMP,d0(dx)
+�
+−
+� 1
+2πi
+�
+Γ
+f(z) Tr G
+�Y (z)dz − 1
+2πi
+�
+Γ
+f(z) Tr GA(0, z)dz
+�
+.
+(C.80)
+Since AA∗ is a usual sample covariance matrix, the first term in the right-hand side converges to a Gaussian
+random variable. Further, as computed in (C.52),
+E[�Y 4
+ij] =: �
+w4
+N 2 +
+1
+(NFg)2
+� ∞
+−∞
+�g′(w)
+g(w)
+�4 �
+g
+�
+w −
+√
+NλuivT
+j
+�
+− g(w)
+�
+dw,
+where the first term is the leading term of E[�Y 4
+ij] and hence the leading term of E[A4
+ij] as well. This means
+that the difference between �w4 and E[A4
+ij] is negligible in the sense that it has no contribution in the limiting
+behavior of the resolvent, which can be checked from standard Green function comparison theorems. (Refer
+to [26].)
+Thus, the mean and the variance of the limiting Gaussian distribution are given by
+mA(f) =
+�f(2) + �f(−2)
+4
+− 1
+2τ0( �f) + (�
+w4 − 3)τ2( �f)
+(C.81)
+and
+VA(f) = 2
+∞
+�
+ℓ=1
+ℓτℓ( �f)2 + (�
+w4 − 3)τ1( �f)2,
+(C.82)
+respectively.
+For the second term in the right-hand side of (C.80), by (C.75), we obtain that
+1
+2πi
+�
+Γ
+f(z) Tr G
+�Y (z)dz − 1
+2πi
+�
+Γ
+f(z) Tr GA(0, z)dz
+= − k
+2πi
+�
+Γ
+f(z) λFg d
+dz(sz + 1)
+λFg(sz + 1) + 1dz + o(1)
+(C.83)
+with high probability. From (C.80), we thus find that the CLT for the LSS holds, i.e.,
+� M
+�
+i=1
+f(�µi) − M
+� d+
+d−
+f(x) ρMP,d0(dx)
+�
+→ N(m�Y (f), V�Y (f)),
+(C.84)
+and the variance V�Y (f) = VA(f) since the second term in (C.80) converges to a deterministic value as
+70
+
+N → ∞, which corresponds to the change of the mean. In particular,
+m�Y (f) − mA(f) = (GH − Fg)λk
+2πi
+�
+Γ
+f(z) ∂
+∂z (zs(z) + 1)dz + k
+2πi
+�
+Γ
+f(z) λFg d
+dz(sz + 1)
+λFg(sz + 1) + 1dz.
+(C.85)
+Following the computation in the proof of Lemma 4.4 in [9] with the relation (C.28), we find that the
+right-hand side of (C.85) is given by
+k
+2πi
+�
+Γ
+f(z)(zs(z) + 1)′
+�
+λ(GH − Fg) +
+λFg
+λFg(zs(z) + 1) + 1
+�
+dz
+= λk
+√d0
+(GH − Fg)τ1( �f) + k
+∞
+�
+ℓ=1
+� λFg
+√d0
+�ℓ
+τℓ( �f).
+(C.86)
+(See also Remark 1.7 of [9].) Thus,
+m��Y (f) =
+�f(2) + �f(−2)
+4
+− 1
+2τ0( �f) + λk
+√d0
+(GH − Fg)τ1( �f) + (�
+w4 − 3)τ2( �f) + k
+∞
+�
+ℓ=1
+� λFg
+√d0
+�ℓ
+τℓ( �f)
+(C.87)
+and
+V�Y (f) = 2
+∞
+�
+ℓ=1
+ℓτℓ( �f)2 + (�
+w4 − 3)τ1( �f)2.
+(C.88)
+C.3
+Proof of Lemma C.7
+Notational remarks
+In the rest of the section, we use C order to denote a constant that is independent of N. Even if the constant
+is different from one place to another, we may use the same notation C as long as it does not depend
+on N for the convenience of the presentation. Now, we recall the linearization HA(θ)(z) and its inverse
+RA(θ, z) = HA(θ)(z)−1. For simplicity, we drop the subscript A and index z of the linearization entries.
+Proof of Lemma C.7. To prove the lemma, we consider
+∂
+∂θ Tr GA(θ, z) =
+�
+b
+�
+a
+�
+α
+∂Aaα(θ)
+∂θ
+∂Gbb(θ)
+∂Aaα(θ)
+=
+�
+b
+�
+a
+�
+α
+∂Haα(θ)
+∂θ
+∂Rbb(θ)
+∂Haα(θ)
+= −
+�
+b
+�
+a
+�
+α
+∂Haα(θ)
+∂θ
+[Rba(θ)Rαb(θ) + Rbα(θ)Rab(θ)]
+= −2
+�
+a
+�
+α
+∂Haα(θ)
+∂θ
+(R(θ)2)aα
+= −2
+�
+a
+�
+α
+∂Haα(θ)
+∂θ
+∂
+∂z Raα(θ),
+(C.89)
+where we again used that
+∂
+∂zGA(θ, z) = GA(θ, z)2. We expand the right-hand side by using the definition of
+71
+
+A(θ),
+Haα(θ) = Aaα(θ) =
+�
+1 − θ + θ
+�
+NSaα
+�
+Aaα(0) =
+�
+1 − θ + θ
+�
+NSaα
+�
+Haα(0),
+(C.90)
+and so
+�
+a
+�
+α
+∂Haα(θ)
+∂θ
+Raα(θ) =
+�
+a
+�
+α
+�
+−1 +
+�
+NSaα
+�
+Haα(0)Raα(θ)
+=
+�
+a
+�
+α
+−1 + √NSaα
+1 − θ + θ√NSaα
+Haα(θ)Raα(θ)
+= Nλ(GH − Fg)
+2
+�
+a
+�
+α
+(uavT
+α)2Haα(θ)Raα(θ) + O(N∥U∥2
+∞∥V ∥2
+∞).
+(C.91)
+From now, we further drop the θ-dependency for the brevity.
+Then
+∂
+∂θ Tr GA(θ, z) = −Nλ(GH − Fg) ∂
+∂z
+�
+a
+�
+α
+(uavT
+α)2HaαRaα + O(N∥U∥2
+∞∥V ∥2
+∞).
+Here, we used the properties that Haα = Aab(θ) = O(N − 1
+2 ), Rab = GA
+ba(θ) = O(N − 1
+2 ) for b ̸= a, Raa =
+GA
+aa(θ) = O(1), and �
+a ua(ℓ1)ua(ℓ2) = δℓ1ℓ2 = �
+α vα(ℓ1)vα(ℓ2), which imply
+�����N 2 �
+a
+�
+α
+(uavT
+α)4HaαRaα
+����� ≤ N 2∥U∥2
+∞∥V ∥2
+∞
+�
+a
+�
+α
+(uavT
+α)2|HaαRaα| = O(N∥U∥2
+∞∥V ∥2
+∞).
+(C.92)
+Together with Remark C.3, from the elementary equality for R and H, we have
+�
+a
+�
+α
+ua(ℓ)2HaαRaα =
+�
+a
+ua(ℓ)2
+��
+α
+HaαRaα
+�
+=
+�
+a
+ua(ℓ)2(1 + zRaa)
+= 1 + zs(z) + O(N − 1
+2 ),
+(C.93)
+and
+�
+a
+�
+α
+vα(ℓ)2HaαRaα =
+�
+α
+vα(ℓ)2
+��
+a
+HaαRaα
+�
+=
+�
+α
+vα(ℓ)2(1 + Rαα)
+= d0(1 + zs(z)) + O(N − 1
+2 ).
+(C.94)
+72
+
+Plugging them into (C.91), we get
+4
+λ(GH − Fg) × (C.91)
+= N
+�
+ℓ
+�
+a
+�
+α
+� 1
+N ua(ℓ)2HaαRaα + 1
+M vα(ℓ)2HaαRaα
++
+�
+ua(ℓ)2 − 1
+M
+�
+vα(ℓ)2HaαRaα + ua(ℓ)2
+�
+vα(ℓ)2 − 1
+N
+�
+HaαRaα
+�
++ 2N
+�
+ℓ1̸=ℓ2
+�
+a
+�
+α
+ua(ℓ1)ua(ℓ2)vα(ℓ1)vα(ℓ2)HaαRaα + O(N∥U∥2
+∞∥V ∥2
+∞)
+= N
+�
+ℓ
+�
+a
+�
+α
+� �
+ua(ℓ)2 − 1
+M
+�
+vα(ℓ)2HaαRaα + ua(ℓ)2
+�
+vα(ℓ)2 − 1
+N
+�
+HaαRaα
+�
++ 2N
+�
+ℓ1̸=ℓ2
+�
+a
+�
+α
+ua(ℓ1)ua(ℓ2)vα(ℓ1)vα(ℓ2)HaαRaα
++ 2k(zs(z) + 1) + O(N∥U∥2
+∞∥V ∥2
+∞).
+(C.95)
+It remains to estimate the first three terms in (C.95). Set
+X1 ≡ X1(θ, z, ℓ) :=
+�
+a
+�
+α
+�
+ua(ℓ)2 − 1
+M
+�
+vα(ℓ)2HaαRaα,
+(C.96)
+X2 ≡ X2(θ, z, ℓ) :=
+�
+a
+�
+α
+ua(ℓ)2
+�
+vα(ℓ)2 − 1
+N
+�
+HaαRaα
+(C.97)
+and
+X3 ≡ X3(θ, z, ℓ1, ℓ2) :=
+�
+a
+�
+α
+ua(ℓ1)ua(ℓ2)vα(ℓ1)vα(ℓ2)HaαRaα
+(ℓ1 ̸= ℓ2).
+(C.98)
+We notice that |X1|, |X2|, |X3| = O(N −1) on Γ1/2 by a naive power counting as in (C.91) after applying
+H¨older inequality once. To obtain a better bound, we use a method based on a recursive moment estimate,
+introduced in [39]. We need the following lemma:
+Lemma C.9. Let X1, X2 and X3 be as in (C.96), (C.97) and (C.98). Define an event Ωε by
+Ωε =
+�
+a,b,α,β
+{|Haα|, |Raα| ≤ N − 1
+2 +ε} ∩ {|Rab − s(z)δab| ≤ N − 1
+2 +ε} ∩ {|Rαβ − zs(z)δαβ| ≤ N − 1
+2 +ε}
+Then, for any fixed (large) D and (small) ε, which may depend on D,
+E[|X|2D|Ωε] ≤ CN − 1
+2 +ε∥u∥2
+∞E[|X|2D−1|Ωε] + CN −1+4ε∥u∥4
+∞E[|X|2D−2|Ωε]
++ CN −2+10ε∥u∥6
+∞E[|X|2D−3|Ωε] + CN −3+14ε∥u∥8
+∞E[|X|2D−4|Ωε],
+(C.99)
+where X is X1, X2 and X3.
+Since the rank of the signal k is fixed, we suffices to prove the above lemma for fixed ℓ, ℓ1 and ℓ2. We will
+prove Lemma C.9 for X1 at the end of this section (the calculation for the X2 and X3 is almost the same).
+With Lemma C.9, we are ready to obtain an improved bound for X. First, note that the contribution from
+the exceptional event Ωc
+ε is negligible i.e., P(Ωc
+ε) < N −D2, which can be checked by applying a high-order
+73
+
+Markov inequality with the moment condition on �Y (See Assumption 3.1). We decompose E[|X|2D] by
+E[|X|2D] = E[|X|2D · 1(Ωε)] + E[|X|2D · 1(Ωc
+ε)] = E[|X|2D|Ωε] · P(Ωε) + E[|X|2D · 1(Ωc
+ε)].
+(C.100)
+Then the second term in the right-hand side of (C.100),
+E[|X|2D · 1(Ωc
+ε)] ≤
+�
+E[|X|4D]
+� 1
+2 (P(Ωc
+ε))
+1
+2 ≤ N − D2
+2 �
+E[|X|4D]
+� 1
+2
+(C.101)
+and by using a trivial bound for the resolvent |Rab(z)| ≤ ∥GA(z)∥ ≤
+1
+Im z
+E[|X|4D] ≤ E
+��
+a
+�
+α
+|HaαRaα|
+�4D
+≤ (M 2N)4D
+(Im z)4D max
+a,b,α E|HaαHbα|4D ≤ CN 14D.
+(C.102)
+To bound the right-hand side of (C.99), we use Young’s inequality: For any a, b > 0 and p, q > 0 with
+1
+p + 1
+q = 1,
+ab ≤ ap
+p + bq
+q .
+We then find that the first term has the following upper bound
+N − 1
+2 +ε∥u∥2
+∞|X|2D−1 = N
+(2D−1)ε
+2D
+N − 1
+2 +ε∥u∥2
+∞ · N − (2D−1)ε
+2D
+|X|2D−1
+≤
+1
+2DN (2D−1)ε(N − 1
+2 +ε∥u∥2
+∞)2D + 2D − 1
+2D
+N −ε|X|2D.
+(C.103)
+Applying Young’s inequality for other terms in (C.99), we get
+E[|X|2D|Ωε] ≤ CN (2D−1)ε(N − 1
+2 +ε∥u∥2
+∞)2D + CN (D−1)ε(N −1+4ε∥u∥4
+∞)D
++ CN ( 2D
+3 −1)ε(N −2+10ε∥u∥6
+∞)
+2D
+3 + CN ( D
+2 −1)ε(N −3+14ε∥u∥8
+∞)
+D
+2
++ CN −εE[|X|2D|Ωε].
+(C.104)
+Absorbing the last term in the right-hand side to the left-hand side and plugging the estimates (C.101) and
+(C.102) into (C.100), we now get
+E[|X|2D] ≤ CN (2D−1)ε(N − 1
+2 +ε∥u∥2
+∞)2D + CN (D−1)ε(N −1+4ε∥u∥4
+∞)D
++ CN ( 2D
+3 −1)ε(N −2+10ε∥u∥6
+∞)
+2D
+3 + CN ( D
+2 −1)ε(N −3+14ε∥u∥8
+∞)
+D
+2 + CN − D2
+2 +7D.
+(C.105)
+From the (2D)-th order Markov inequality, for any fixed ε′ > 0 independent of D,
+P
+�
+|X| ≥ N ε′N − 1
+2 ∥u∥2
+∞
+�
+≤ N −2Dε′
+E[|X|2D]
+(N − 1
+2 ∥u∥2∞)2D ≤ N −2Dε′N 8Dε.
+(C.106)
+By choosing ε = 1/D, for sufficiently large D, we find that
+|X| = O(N − 1
+2 ∥u∥2
+∞).
+(C.107)
+74
+
+We now return to (C.89) and use (C.95) with the bound (C.107),
+M
+�
+j=1
+N
+�
+k=1
+∂Ajk(θ)
+∂θ
+(GA(θ)A(θ))jk = (GH − Fg)λk
+2
+(1 + zs(z)) + O(N∥u∥2
+∞∥v∥2
+∞).
+(C.108)
+To handle the derivative of the right-hand side, we use Cauchy’s integral formula with a rectangular contour,
+contained in Γε
+1/2, whose perimeter is larger than ε. Then, we get from (C.89) that
+∂
+∂θ Tr GA(θ, z) = −λ(GH − Fg) · k ∂
+∂z (1 + zs(z)) + O(N∥u∥2
+∞∥v∥2
+∞).
+(C.109)
+After integrating over θ from 0 to 1, we conclude that (C.70) holds for a fixed z ∈ Γε
+1/2.
+At last, we prove Lemma C.9.
+Proof of Lemma C.9. As we mentioned above, we consider X = X1 and drop the ℓ-dependency. i.e.
+E[|X|2D] = E
+��
+a
+�
+α
+�
+u2
+a − 1
+M
+�
+v2
+αHaαRaαXD−1X
+D
+�
+We use the following inequality that generalizes Stein’s lemma (see Proposition 5.2 of [11]): Let Φ be a C2
+function. Fix a (small) ε > 0, which may depend on D. Recall that Ωε is the complement of the exceptional
+event on which |Haα| or |Raα| is exceptionally large for some a, α, defined by Ωε by
+�
+a,b,α,β
+{|Haα|, |Raα| ≤ N − 1
+2 +ε} ∩ {|Rab − s(z)δab| ≤ N − 1
+2 +ε} ∩ {|Rαβ − zs(z)δαβ| ≤ N − 1
+2 +ε}
+Then,
+E[HaαΦ(Haα)|Ωε] = (E[H2
+aα|Ωε] − E[Haα|Ωε]2)E[Φ′(Haα)|Ωε] + ε1,
+(C.110)
+where the error term ε1 admits the bound
+|ε1| ≤ C1E
+�
+|Haα|3 sup
+|t|≤1
+Φ′′(tHaα)
+���Ωε
+�
+(C.111)
+for some constant C1. Note that by applying a decomposition (C.100) to E[Haα|Ωε] and E[H2
+aα|Ωε], we see
+that
+E[Haα|Ωε] − E(E[Haα|Ωε]) = E[Haα|Ωε] = O(N −D0)
+(C.112)
+and
+E[H2
+aα|Ωε] = E(E[H2
+aα|Ωε]) + O(N −D0) = 1
+N + O(∥u∥2
+∞∥v∥2
+∞) + O(N −D0)
+(C.113)
+for D0 = D2+1
+2
+> 1. The estimate (C.110) follows from the proof of Proposition 5.2 of [11] with p = 1, where
+we use the inequality (5.38) therein only up to second to the last line.
+In the estimate (C.110), we choose
+Φ(Haα) = RaαXD−1X
+D
+(C.114)
+so that
+E[|X|2D|Ωε] =
+�
+a
+�
+α
+�
+u2
+a − 1
+M
+�
+v2
+αE [HaαΦ(Haα)|Ωε] .
+(C.115)
+75
+
+Applying (C.112) and (C.113) to the equation (C.110),
+E [HaαΦ(Haα)|Ωε] = E
+�
+H2
+aα
+�
+E[Φ′(Haα)|Ωε] + ε1
+= E[H2
+aα]
+�
+−E
+�
+RaaRααXD−1X
+D|Ωε
+�
+− E
+�
+R2
+aαXD−1X
+D|Ωε
+�
++(D − 1)E
+�
+Raα
+∂X
+∂Haα
+XD−2X
+D��Ωε
+�
++ DE
+�
+Raα
+∂X
+∂Haα
+XD−1X
+D−1��Ωε
+��
++ ε1,
+(C.116)
+for sufficiently large D. We plug it into (C.115) and estimate each term. Then the term originated from the
+first term in (C.116) can be separated by
+�
+a
+�
+α
+�
+u2
+a − 1
+M
+�
+v2
+αE[H2
+aα]E
+�
+RaaRααXD−1X
+D|Ωε
+�
+=
+�
+a
+�
+α
+�
+u2
+a − 1
+M
+�
+v2
+αE[H2
+aα]E
+�
+(Raa − s)RααXD−1X
+D|Ωε
+�
++ s
+�
+a
+�
+α
+�
+u2
+a − 1
+M
+�
+v2
+αE[H2
+aα]E
+�
+RααXD��1X
+D|Ωε
+�
+.
+(C.117)
+The first term satisfies that
+�����
+�
+a
+�
+α
+�
+u2
+a − 1
+M
+�
+v2
+αE[H2
+aα]E
+�
+(Raa − s)RααXD−1X
+D|Ωε
+������
+≤ CM∥u∥2
+∞N −1N − 1
+2 +εE[|X|2D−1|Ωε]
+�
+α
+v2
+α = CN − 1
+2 +ε∥u∥2
+∞E[|X|2D−1|Ωε]
+(C.118)
+for some constant C since �
+α v2
+α = 1. Using (C.113) and �
+a
+�
+u2
+a −
+1
+M
+�
+= 0, we also have
+�����s
+�
+a
+�
+α
+�
+u2
+a − 1
+M
+�
+v2
+αE[H2
+aα|Ωε]E
+�
+RααXD−1X
+D|Ωε
+������
+≤ C∥u∥2
+∞∥v∥2
+∞|s|
+�
+a
+�
+α
+�
+u2
+a + 1
+M
+�
+v2
+αE
+�
+|RααXD−1X
+D||Ωε
+�
+≤ C∥u∥2
+∞∥v∥2
+∞E[|X|2D−1|Ωε]
+(C.119)
+for some constant C and large D > 1. For the second term in (C.116), we also have
+�����
+�
+a
+�
+α
+�
+u2
+a − 1
+M
+�
+v2
+αE[H2
+aα|Ωε]E
+�
+R2
+aαXD−1X
+D|Ωε
+������
+≤ CN −1∥u∥2
+∞
+�����
+�
+a
+�
+α
+v2
+αE
+�
+R2
+aαXD−1X
+D|Ωε
+������
+≤ CN −1+2ε∥u∥2
+∞E[|X|2D−1|Ωε]
+�
+α
+v2
+α
+≤ CN −1+2ε∥u∥2
+∞E[|X|2D−1|Ωε].
+(C.120)
+76
+
+To estimate the third term and the fourth term in (C.116), we notice that on Ωε
+����
+∂X
+∂Haα
+���� =
+������
+−
+�
+b
+�
+β
+�
+u2
+b − 1
+M
+�
+v2
+βHbβ[RabRαβ + RbαRaβ] +
+�
+u2
+a − 1
+M
+�
+v2
+αRaα
+������
+≤ CN − 1
+2 +3ε∥u∥2
+∞
+�
+α
+v2
+α + CN − 1
+2 +ε∥u∥2
+∞∥v∥2
+∞ ≤ CN − 1
+2 +3ε∥u∥2
+∞.
+(C.121)
+for some constant C. Similarly, we can observe that
+����
+∂2X
+∂H2aα
+���� ≤ CN − 1
+2 +3ε∥u∥2
+∞.
+(C.122)
+Thus, we also obtain that
+�����
+�
+a
+�
+α
+�
+u2
+a − 1
+M
+�
+v2
+αE[H2
+aα|Ωε]E
+�
+Raα
+∂X
+∂Haα
+XD−2X
+D��Ωε
+������
+≤ CN −1+4ε∥u∥4
+∞E[|X|2D−2|Ωε]
+(C.123)
+and
+�����
+�
+a
+�
+α
+�
+u2
+a − 1
+M
+�
+v2
+αE[H2
+aα|Ωε]E
+�
+Raα
+∂X
+∂Haα
+XD−1X
+D−1��Ωε
+������
+≤ CN −1+4ε∥u∥4
+∞E[|X|2D−2|Ωε].
+(C.124)
+Hence, from (C.116), (C.120), (C.123), and (C.124),
+�����
+�
+a
+�
+α
+�
+u2
+a − 1
+M
+�
+v2
+αE[H2
+aα|Ωε]E[Φ′(Haα)|Ωε]
+�����
+≤ CN − 1
+2 +ε∥u∥2
+∞E[|X|2D−1|Ωε] + CN −1+4ε∥u∥4
+∞E[|X|2D−2|Ωε] + ε1.
+(C.125)
+It remains to estimate |ε1| in (C.111). Proceeding as before,
+�
+a
+�
+α
+�
+u2
+a − 1
+M
+�
+v2
+αE
+�
+|Haα|3Φ′′(Haα)
+���Ωε
+�
+≤ CN −1+4ε∥u∥2
+∞E[|X|2D−1|Ωε] + CN −2+7ε∥u∥4
+∞E[|X|2D−2|Ωε]
++ CN −2+10ε∥u∥6
+∞E[|X|2D−3|Ωε].
+(C.126)
+Our last goal is to find the bound for the error term ε1. To handle Φ′′(tHaα), we want to compare
+Φ′′(Haα) and Φ′′(tHaα) for some |t| < 1. Let GA,t be the resolvent of A where Aaα is replaced by tAaα,
+and let Xt be defined as X in (C.96) with the same replacement for Aaα and also GA is replaced by GA,t.
+Correspondingly, we also consider the replacement Rt of the linearization R by substituting tHaα into Haα
+(also for Hαa). Then,
+Rt
+AB − RAB = (Rt(H − Ht)R)AB = (1 − t)Rt
+AaHaαRαB + (1 − t)Rt
+AαHαaRaB.
+(C.127)
+77
+
+and
+Xt − X =
+�
+b
+�
+β
+�
+u2
+b − 1
+M
+�
+v2
+β(Ht
+bβRt
+bβ − HbβRbβ)
+=
+�
+b
+�
+β
+�
+u2
+b − 1
+M
+�
+v2
+βHbβ(Rt
+bβ − Rbβ) + (t − 1)
+�
+u2
+a − 1
+M
+�
+v2
+α(HaαRt
+aα)
+= (1 − t)
+�
+b
+�
+β
+�
+u2
+b − 1
+M
+�
+v2
+βHbβRt
+baHaαRαβ
++ (1 − t)
+�
+b
+�
+β
+�
+u2
+b − 1
+M
+�
+v2
+βHbβRt
+bαHαaRaβ + (t − 1)
+�
+u2
+a − 1
+M
+�
+v2
+αHaαRt
+aα.
+(C.128)
+Thus, on Ωε,
+|Xt − X| ≤ CN −1+4ε∥u∥2
+∞.
+(C.129)
+Using the estimates (C.127) and (C.129), on Ωε, we obtain that
+|Φ′′(Haα) − Φ′′(tHaα)| ≤ C|Φ′′(Haα)| + N − 5
+2 +11ε∥u∥6
+∞|X|2D−4
+(C.130)
+uniformly on t ∈ (−1, 1).
+Combining (C.115) and (C.125) with (C.126), (C.130), and (C.111), we finally get
+E[|X|2D|Ωε] ≤ CN − 1
+2 +ε∥u∥2
+∞E[|X|2D−1|Ωε] + CN −1+4ε∥u∥4
+∞E[|X|2D−2|Ωε]
++ CN −2+10ε∥u∥6
+∞E[|X|2D−3|Ωε] + CN −3+14ε∥u∥8
+∞E[|X|2D−4|Ωε].
+(C.131)
+This proves the desired lemma for X = X1.
+For the cases X = X2 or X = X3, the proofs are almost the same with the following changes:
+• For X = X2, we change the role of U and V . In other words, we will use �
+a ua(ℓ)2 = 1 and
+�
+α
+�
+vα(ℓ)2 − 1
+N
+�
+= 0. Then the recursive bound for E[|X2|2D|Ωε] obtained by putting v instead of
+u in the upper bound in (C.99).
+• In the same way, we use �
+a ua(ℓ1)ua(ℓ2) = δℓ1ℓ2 and �
+α |vα(ℓ1)||vα(ℓ2)| ≤ 1 instead of �
+a
+�
+u2
+a −
+1
+M
+�
+=
+0 and �
+a ua(ℓ)2 = 1, respectively. We then obtain the exactly same recursive bound in (C.99) for X3.
+C.4
+Computation of the test statistic
+In this section, we prove the second part of Theorem 5.5 and also provide the details on the computation of
+the test statistic in Theorem 4.2. By performing the same calculations as we will do in this section, we can
+obtain optimal functions for the other models, so we omit the details. (Refer to [22, 33, 34].) Recall that
+mY (f)|H1 − mY (f)|H0 =
+k
+�
+s=1
+∞
+�
+ℓ=1
+� ωs
+√d0
+�ℓ
+τℓ( �f)
+(C.132)
+and
+VY (f) = 2
+∞
+�
+ℓ=2
+ℓτℓ( �f)2 + (w4 − 1)τ1( �f)2.
+(C.133)
+78
+
+Assuming w2 > 0 and w4 > 1, from Cauchy’s inequality and the identity log(1 − λ) = − �∞
+ℓ=1 λℓ/ℓ,
+�����
+mY (f)|H1 − mY (f)|H0
+�
+VY (f)
+�����
+2
+≤
+k
+�
+p,q=1
+ωpωq
+d0
+�
+1
+w4 − 1 − 1
+2
+�
+− 1
+2 log
+�
+1 − ωpωq
+d0
+�
+=
+����
+m(Ω) − m(0)
+√V0
+����
+2
+,
+(C.134)
+which proves the first part of the theorem. The equality in (C.134) holds if and only if
+√d0(w4 − 1)τ1( �f)
+�
+s ωs
+= 2ℓ(√d0)ℓτℓ( �f)
+�
+s ωℓs
+(ℓ = 2, 3, 4, . . . ).
+(C.135)
+We now find all functions f that satisfy (C.135). Letting 2C be the common value in (C.135),
+τ1( �f) =
+2C
+√d0(w4 − 1)
+�
+s
+ωs,
+τℓ( �f) =
+C
+ℓ(√d0)ℓ
+�
+s
+ωℓ
+s
+(ℓ = 2, 3, 4, . . . ).
+(C.136)
+We can expand �f in terms of the Chebyshev polynomials as
+�f(x) =
+∞
+�
+ℓ=0
+CℓTℓ
+�x
+2
+�
+.
+(C.137)
+The orthogonality relation of the Chebyshev polynomials implies that for ℓ ≥ 1
+τℓ( �f) = Cℓ
+π
+� 2
+−2
+Tℓ
+�x
+2
+�
+Tℓ
+�x
+2
+�
+dx
+√
+4 − x2 = Cℓ
+π
+� 1
+−1
+Tℓ (y) Tℓ (y)
+dy
+�
+1 − y2 = Cℓ
+2 .
+(C.138)
+Thus, (C.136) holds if and only if
+�f(x) = c0 + 2C
+�
+s
+�
+2ωs
+√d0(w4 − 1)T1
+�x
+2
+�
++
+∞
+�
+ℓ=2
+1
+ℓ
+� ωs
+√d0
+�ℓ
+Tℓ
+�x
+2
+��
+= c0 + 2C
+�
+s
+�
+ωs
+√d0
+�
+2
+w4 − 1 − 1
+�
+T1
+�x
+2
+�
++
+∞
+�
+ℓ=1
+1
+ℓ
+� ωs
+√d0
+�ℓ
+Tℓ
+�x
+2
+��
+(C.139)
+for some constant c0. We notice that the following identity holds for the Chebyshev polynomials:
+∞
+�
+ℓ=1
+tℓ
+ℓ Tℓ (x) = log
+�
+1
+√
+1 − 2tx + t2
+�
+.
+(C.140)
+(See, e.g., (18.12.9) of [45].) Since T1(x) = x, we find that (C.139) is equivalent to
+�f(x) = c0 + C
+�
+s
+� ωs
+√d0
+�
+2
+w4 − 1 − 1
+�
+x − log
+�d0 − ωs
+√d0x + ω2
+s
+d0
+��
+,
+(C.141)
+79
+
+or
+f(x) = c0 + C
+�
+s
+�ωs
+d0
+�
+2
+w4 − 1 − 1
+�
+x − ωs(1 + d0)
+d0
+�
+2
+w4 − 1 − 1
+��
+− C
+�
+s
+log
+�ωs
+d0
+��
+1 + d0
+ωs
+�
+(1 + ωs) − x
+��
+.
+(C.142)
+This concludes the proof of Theorem 5.2 with an optimal function
+φΩ(x) = �φΩ(ϕ(x))
+(C.143)
+where
+�φΩ(x) = c0 +
+�
+s
+� ωs
+√d0
+�
+2
+w4 − 1 − 1
+�
+x − log
+�d0 − ωs
+√d0x + ω2
+s
+d0
+��
+.
+(C.144)
+Choosing
+c0 =
+�
+s
+�(1 + d0)
+d0
+�
+2
+w4 − 1 − 1
+�
+ωs + log(ωs/d0)
+�
+,
+we get (4.2). Further, we can see that
+φΩ(x) =
+�
+s
+φωs(x).
+(C.145)
+From this, we directly obtain that LΩ = �
+s Lωs,
+mY (φω)|H0 = −1
+2
+�
+s
+log
+�
+1 − ω2
+s
+d0
+�
++
+1
+2d0
+(w4 − 3)
+�
+s
+ω2
+s,
+(C.146)
+mY (φω)|H1 = mY (φω)|H0 +
+�
+p,q
+�
+− log
+�
+1 − ωpωq
+d0
+�
++ ωpωq
+d0
+�
+2
+w4 − 1 − 1
+��
+(C.147)
+and
+VY (φω)|H1 = VY (φω)|H0 = 2
+�
+p,q
+�
+− log
+�
+1 − ωpωq
+d0
+�
++ ωpωq
+d0
+�
+2
+w4 − 1 − 1
+��
+.
+(C.148)
+80
+