Fluctuations of the largest eigenvalues of transformed spiked Wigner matrices
Authors: Aro Lee
Ji Oon Lee
Venue: ICML 2025 poster (2025)
Keywords: spiked random matrix
Wigner matrix
largest eigenvalue
PCA
signal detection
Primary area: theory->learning_theory
TL;DR: We prove the fluctuation of the largest eigenvalue of a spiked random matrix model, obtained by applying a function entrywise to a signal-plus-noise symmetric data matrix.

Abstract: We consider a spiked random matrix model obtained by applying a function entrywise to a signal-plus-noise symmetric data matrix. We prove that the largest eigenvalue of this model, which we call a transformed spiked Wigner matrix, exhibits Baik-Ben Arous-Péché (BBP) type phase transition. We show that the law of the fluctuation converges to the Gaussian distribution when the effective signal-to-noise ratio (SNR) is above the critical number, and to the GOE Tracy-Widom distribution when the effective SNR is below the critical number. We provide precise formulas for the limiting distributions and also concentration estimates for the largest eigenvalues, both in the supercritical and the subcritical regimes.

Fluctuations of the Largest Eigenvalues of Transformed Spiked Wigner Matrices
Aro Lee
Ji Oon Lee
Abstract
with first layer weights. Additionally, these models have practical applications in machine learning algorithms, such as an ensemble method for Q-learning reinforcement learning (Lee et al., 2023a).
We consider a spiked random matrix model ob- tained by applying a function entrywise to a signal-plus-nois symmetric data matrix. We prove that the largest eigenvalue of this model, which we call transformed spiked Wigner ma- trix, exhibits phase transition. We show that the law of the fluctuation converges to the Gaussian distribution when the effective signal-to-noise ratio (SNR) is above the critical number, and to the GOE Tracy- Widom distribution when the effective SNR below the critical number. We provide precise formulas for the limiting distributions and also concentration estimates for the largest eigenval- ues, both in the supercritical and the subcritical regimes.
One of the most natural models for signal plus-noise data spiked Wigner matrix, which is of the form
Arous Péché (BBP) type
M = W + Vxxe"
where the noise W is N N symmetric (real) Wigner ma- trix, x: RN is the spike, and À E IR is the hignal-tonnoiss ratio (SNR). (See Definitions 2.1 and 2.2.) Spiked Wigner matrices appear naturally in inference problems where the data is obtained from pairwise measurements. Notable ex- amples of such problems include the community detection from the stochastic block model and synchronization over Z/2.
It is well-known that the largest eigenvalue of M exhibits phase transition, known as the BBP transition named after the seminal work by Baik, Ben Arous, and Péché (Baik et al., 2005), depending on the parameter A. If > 1, then the largest eigenvalue of A converges to V vI separated from the other eigenvalues of M. On the other hand, if X < 1, the largest eigenvalue converges to 2, which the spectral edge of the pure noise model.
1. Introduction
First introduced in the early 1900s, the principal compo- nent analysis (PCA) has been widely used as fundamental method for analyzing multivariate data across wide range of scientific fields including genetics, economics, and var- ious other disciplines. One notable application of PCA is detecting and recovering a signal from matrix-type data that includes inevitable noise, commonly referred to signal- plus-noise data.
In this paper, we consider an entrywise transformation of the spiked Wigner matrix (1), defined by
When the signal is low-rank matrix, the signal-plus-noiss. data can be modeled by spiked random matrix models, which play a crucial role in analyzing for many machine learning problems. For instance, in the study of the feature learning a two-layer neural network, rank-1 perturbation of the initial weight matrix can approximate the updated weight after one gradient step (Ba al., 2022). This idea is further developed by (Cui et al., 2024; Dandi et al., 2025), where the spiked Random Features model was considered
My N -
(2)
for function f. (Note that typical size of the entries of a Wigner matrix of order N-1/2.) Such an entrywise transformation was considered in (Lesieur al., 2015; Perry et al., 2018) improve PCA by effectively increasing the SNR. Heuristically, the entries of the transformed spiked Wigner matrix in (2) can be approximated using the Taylor expansion by
Department of Mathematical Sciences, KAIST, Daejeon, Ko- rea. Correspondence to: Aro Lee <sditar444 @ kr.>t, Ji Oon Lee <jjoon. leroon.lee@aaii @aiist.euu....
N J( ((NNM,) = N N + VANNYR (VNW,,)
+ +VAF(VNW.)DOR)
VN
(3)
Proceedings of the ndnd International Conference Machine Learning. Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).
f(vNW
+
VN

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Largest Eigenvalues of Transformed Spiked Wigner Matrices
The transformed matrix M is approximately spiked Wigner matrix; where the SNR is changed to (V )))². By optimizing f. it is possible to make the new SNR larger than the original SNR, which makes the PCA with the transformed matrix (called the transformed PCA) stronger than the vanilla PCA. The approximation in (3) is proven to be valid in (Perry et al., 2018), in the sense that the difference between the largest eigenvalues of M and its approximation defined by the right side of (3) is o(1). (Perry et al., 2018) considers the approximation in (3) only for the optimal transformation, the same proof works for any transformation.
M. We also consider some specific examples to compare the numerical results with the theoretical results.
The main technical difficulty in the proof is that the error terms in the approximation in (3) is not negligible priori. Most precisely, in (3), the term
V
- -IF())))FFWWWW).. 2,z,
(4)
is ignored. However, if we follow the analysis in (Perry et al.. 2018), the norm of the matrix whose entries are given by (4) is of order N-1/2 which is of the same size as the fluctuation of the largest eigenvalue of a spiked Wigner ma- trix in the supereritical case, and much larger than that of the largest eigenvalue in the suberitical case. It is thus im- possible to ignore the term in (4), but then the noise matrix is no more a Wigner matrix since the size of x; and z; may not be uniform. Another issue is that the second derivative term in the Taylor expansion in (3) is not negligible, since it is also of order N 1/2 a priori. Thus, the correct approx- imation should be not by a (rank-1) spiked Wigner matrix but rank-2 spiked Wigner-type matrix. (See Proposition 3.1 for more detail.)
). We note that, while Theorem 4.8 in
Suppose now that we want to detect the presence of the signal in the given data matrix by PCA. It would require to compute the largest eigenvalues of the data matrix, estimate the p-value from the known limiting distributions of the largest eigenvalues, and compare with a predefined signifi- cance level; for algorithm involving the largest eigenvalue only, see (Kritchman& & Nadler, 2008), and for an algorithm based on the ratio between the top eigenvalues, see (Onatski, 2009). If the given data matrix is spiked Wigner matrix, the limiting distributions of the largest cigenvalues are also well-known, which is also part of the BBP transition; if A >>1 1, then the fluctuation of the largest eigenvalue of M is given by a Gaussian of order N-1/2, whereas if X < 1, it is given by GOE Tracy-Widom distribution of order N 22/3 However, a corresponding result for transformed spiked Wigner matrices are not known.
To prove the BBP transition for the rank-2 spiked Wigner- type matrix, we apply the Green function comparison argu- ment. In this strategy, the distribution function of the largest eigenvalue is first approximated by a functional of the re- solvent of the matrix. Then, by comparing the resolvent of the given matrix and that of the target matrix for which the limiting distribution of the largest eigenvalue is known, it is possible to prove that the fluctuations of the largest eigenvalues of two matrices are equal. The target matrices are a rank-2 spiked Wigner matrix for the supercritical case, and a Wigner matrix for the subcritical case. The Green function comparison argument is conducted by following continuous matrix flow, typically called an interpolation, and it requires a technical input, known the local law, which is the estimate on the resolvent of the matrices.
1.1. Contributions
In this paper, we prove the fluctuations of the largest eigen- value of the transformed spiked Wigner matrix M coincide with those of the spiked Wigner matrix. More precisely, under mild assumptions on M (see Assumptions 2.3 and 2.4) if we let μ1 be the largest eigenvalue of M and the effective SNR A = X(EIf
then
The rigidity of the largest eigenvalue is one of the main technical inputs for the Green function comparison argu- ment, and it also of separate interest since it asserts strong concentration of the largest eigenvalue in near-optimal scale. In the subcritical case, the rigidity is consequence of a stronger estimate on the difference between the largest eigen- values of the noise matrix with and without a spike, known as the eigenvalue sticking.
(Supercritical case) If X₂ > 1, then /1 - (VX.+ Vx_)) converges to a Gaussian distribution with mean 0 and variance 2A) 1)/Xe.
(Suberitical case) If A₂ < 1, then 22/3 2) con- verges to the GOE Tracy-Widom distribution.
(Rigidity) For both the supercritical and the suberitical cases, the deviation of the largest eigenvalue from its limit cannot be significantly larger than the typical size of the fluctuation with overwhelming probability.
1.2. Applications
Our results can provide theoretical background for the under- standing of transformed spiked Wigner matrices, especially (asymptotically) exact error probability of the improved PCA for the finite N case. We list possible examples where transformed spiked Wigner matrices are useful.
See Theorems 2.5 and 2.6 for the precise statements. This establishes the BBP transition for the largest eigenvalue of
2

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Largest Eigenvalues of Transformed Spiked Wigner Matrices
Improved PCA. It was proved in (Perry et al., 2018) that PCA achieves the optimal strong detection threshold for spiked Wigner matrix with Gaussian noise, in the sense that it is impossible to reliably distinguish (with probability 1 - o(1) a Wigner matrix with a spike and one without a spike if X < 1. The threshold for the strong detection is lowered the noise is non-Gaussian, but can be matched by improving PCA via entrywise transformation, with the function f = -p' /p where p is the density of the noise.
(Lesieur al., 2015; Krzakala al., 2016; Dia et al., 2016; Lelarge & Miolane, 2019). Entrywise transformation of spiked Wigner matrices was first described by Lesieur, Krza- kala, and Zdeborová (Lesieur et al., 2015) and rigorously analyzed by Perry, Wein, Bandeira, and Moitra (Perry et al., 2018). The transition of the limit of the largest eigenvalue of transformed spiked Wigner matrices was first proved in (Perry et al., 2018), where the correlation between the top eigenvector and the spike was also proved. Similar re- sults were proved with weaker assumptions (Guionnet et al., 2023; Feldman, 2025; Moniri & Hassani, 2024) and also generalized to other models (Jung et al., 2021; Guionnet et al., 2023; Feldman, 2025; Mergny et al., 2024), including spiked rectangular matrices.
Machine learning theory. In deep neural networks, the transformed spiked Wigner matrix can be used in theoretical analysis, where the noise corresponds to the pre-activation, the spike is due to initial training, and the entrywise trans- formation is the activation function. Another important ex- ample is the feature matrix of the two-layer neural network, where the SNR corresponds to the step size.
The spiked random matrix models and their entry trans- formation have been extensively used in the theoretical study of neural networks. These models contribute to the theo- retical analysis of deep neural network, where entrywise transformations correspond to pointwise nonlinear activa- tion functions (Pennington & Worah, 2017). They are also crucial in the analysis on the feature learning. See, e.g., (Damian al., 2022; Ba et al., 2022; Lee et al., 2023b; al., 2023; Moniri al., 2024; Cui et al., 2024; Ba et al., 2024). For more results, we refer to (Cui et al., 2024) and references therein.
1.3. Related Works
The spiked random matrix model was introduced by John- stone (Johnstone, 2001) for Wishart matrices (Gaussian ii.d. rectangular matrices) with spiked covariance. The transition of the largest eigenvalue was proved by Baik, Ben Arous, and Péché (Baik et al., 2005) for spiked com- plex Wishart matrices and generalized to other models, in- cluding the spiked Wigner matrices (Péché, 2006; Féral & Péché, 2007: Capitaine et al., 2009; Benaych-Georger & Nadakuditi, 2011; 1; Knowles & Yin, 2013; Pizzo et al., 2013; Renfrew & Soshnikov, 2013; Knowles & Yin, 2014; Bloemendal et al., 2016). The detection problems for the spiked Wigner matrices have been extensively studied re- cently, and many important results, including the theoretical limits (Perry et al., 2018; El Alaoui et al., 2020; Chung et al., 2025; Moitra & Wein, 2025) and algorithms (Perry et al., 2018; Chung & Lee, 2019; 2022), have been proved. For more results on the spiked Wigner matrices, we refer to (Moitra& & Wein, 2025) and references therein.
Mousavi-
1.4. Organization of the Paper
The rest of the paper is organized follows: In Section 2, we precisely define the model and state our main results. In Section 3 and Section 4, we outline the proof' of our main results for the supercritical case and the subcritical case, re- spectively. In section 5, we conduct numerical experiments to compare theoretical results with numerical results for several examples. We conclude the paper in Section 6 with summary of our results and directions for future research. Details of the numerical simulations and the proofs of the results can be found in Appendices.
The spectral properties of spiked random matrix models are important in analyzing various problems in machine learning and statistics, as spectral methods can improve convergence analysis and provides theoretical guarantees. Notable examples include the community detection (Abbe, 2017) and submatrix localization (Butucea et al., 2013). In the context of machine learning, we refer to (Mondelli & Venkataramanan. 2021) for an application to the approxi- mate message passing algorithm and (Chi et al., 2019) to non-convex optimization.
Notational Remark. We use the standard big-O and little-o notation. For an event S we say that Ω holds with over- whelming probability if for any (large) D > 0 there exists No = No(D) such that PP(()) < N- whenever N > No- For sequence of random variables, the notation + denotes the convergence in distribution as N + oc.
2. Main Result
2.1. Definition of the Model
Entrywise transformed random matrix models provide valu- able insights into the study of fundamental limits of de- tection for the signal from spiked random matrix, which has been extensively studied by analyzing the eigenvalues (Montanari et al., 2015; Johnstone & Onatski, 2020) or the mutual information and minimum mean squared error
We first define the model we consider in this paper. We assume that the noise matrix is Wigner matrix for which we use the following definition.
Definition 2.1 (Wigner matrix). We say an N x N random matrix W = (W.,) is a Wigner matrix W is symmetric
3

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Largest Eigenvalues of Transformed Spiked Wigner Matrices
and Wi (1 S i S N) are independent real random
For all i and j. B(J()NW,)) )) 0. Furthermore, E[ = and Bf (V 2 0.
variables satisfying the following conditions:
For all i, B[W,,] [n = and for any (N -iddeeppendent
We remark that the last part of Assumption 2.4 is not re- strictive, since it can be satisfied by simply multiplying the function f by a suitable constant.
positive integer D. E// S Cp for some
(N -ndindepend)nt) constant CD.
For all i < j. N) [2] 1.
2.2. Main Result
Note that the upper triangle entries of a Wigner matrix are not necessarily identically distributed. The signal-plus- noise model we consider a (rank-1) spiked Wigner matrix, which is defined follows:
Our main result is the following theorem asymptotic nor- mality of the largest eigenvalue of the transformed matrix. Theorem 2.5. Suppose that M is a spiked Wigner matrix, satisfying Assumplion 2.3. Let M be matrix defined by Mi N for function I satisfying As- sumption 2.4. Let (MM be the largest eigenvalue of M. Set
Definition 2.2 (Spiked Wigner matrix). We say an N x N random matrix M- = W + vixx" is a spiked Wigner matrix with a spike æ and signal-to-noise ratio (SNR) if W is a Wigner matrix and æ (2,,22..... ................................ .2N) E R with | =.
de := (VEI3((/W1122))
(5)
Throughout the paper, we assume that À is independent of N. (See Remark 2.8 for the discussion on the case X >> 1.) In addition, we assume the following for the data matrix. Assumption 2.3. Let M W + Vxææ" be spiked Wigner matrix in Definition 2.2. For the spike (#1, .., EN), we assume the following:
(Supercritical c If de 1, then
N//2 (MI(M)) (Vie+ )= N'(0,0
(6)
where the right-hand side of (6) a Gaussian distri- bution with mean O and variance
For any e > 0, max; Jas = O(N Σ: O(N)), and Σ, x = O(N
2² 2(Ae 1)
2(A(E 12))))
N
X(I)"(VN112)))
For the noise matrix W, we assume that the following:
(Suberitical case) If de < 1, then
For all i S j. the normalized entries VNWy are iden- tically distributed.
N2/3 (M) -2) 2 TW,,
(8)
For all é,j and any fixed D, the D-th moment of VNW,, is finite.
where the right-hand side of (8) is the GOE Tracy- Widom distribution.
We remark that the assumption on the spike is satisfied with the i.i.d. prior, where we let yi be i.i.d. with an N- independent distribution whose mean is O, variance 1, and all moments are finite, and set 2; yı/ /N. The assumption is also satisfied for many other priors such as the spherical prior. We believe our result can also be proved with weaker assumptions. Especially, the same result would hold even when the distribution of the diagonal entries are different from that of the off-diagonal entries. However, we do not pursue it in the current paper.
Theorem 2.5 shows that our model, the transformed spiked Wigner matrix, exhibits the BBP transition, and it coincides with that of the tmon-transformedt spiked Wigner matrix even in terms of the fluctuation with A₂ being the effective SNR.
In addition to Theorem 2.5, we also have the following result on the rigidity of the largest eigenvalue.
Theorem 2.6. Suppose that the assumptions in Theorem 2.5 hold. Then, for any > O, the following holds with overwhelming probability.
Lastly, we assume that the entrywise transformation satisfies the following properties.
(Supercritical case) Xe 1, then
A Assumption 2.4. For a given Wigner matrix W. we assume that the following holds with the function f IR + R.
p1(M) - (VTe+ + ( = OON
(9)
vxv
The function f is C and its derivatives are polyno- mially bounded in the sense that (w)) < Cc(1+ +
(Subcritical case) If Xo < 1, then
[w]) (e = 0, 2,3)) for some constant Ct.
in (M) OON- 3+0).
(10)

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Largest Eigenvalues of Transformed Spiked Wigner Matrices
For the supercritial case, Theorem 2.6 was essentially proved in (Perry al., 2018). For the sake of complete- ness, we will prove it in Appendices. The rigidity of the largest eigenvalue for the suberitical case is a consequence of a stronger result, known as the eigenvalue sticking. For a spiked Wigner matrix in the suberitical case, it means that the difference between the largest eigenvalue of a Wigner matrix W and that of the spiked Wigner matrix W+V +xxx" is much less than N-2/3 which is the size of the fluctua- tion of the largest eigenvalue. (See, e.g., Theorem 2.7 in (Knowles & Yin, 2013).)
where x² (1 2z ........ xx) Notice that for the i.i.d. prior, ||a>21| NN + o(N-
In (Guionnet et al., 2023), it was proved that the limit of the largest eigenvalue exhibits the BBP transition with the effective SNR Xe- Adapting the strategy for the proof of Theorems 2.5 and 2.6, it is also possible to prove that the corresponding statements for the fluctuation of the largest eigenvalue. We remark that the actual statement also con- tains a deterministic shift term of order N-1/22 see (79) for more precise statement. In Appendix D, we provide the idea of the proof for this case. It can be further generalized by considering
Remark 2.7. Adapting the strategy for the proof of Theo- rems 2.5 and 2.6, we believe that it is also possible to prove corresponding statements for transformed rank-k spiked Wigner matrices for any fixed k 2 2. A rank-k spiked Wigner matrix is of the form
ky = inffk E Z+ :EI( +
* 0}
and the scaling X = N2 m See Appendix D.4 for more discussion.
M W+
Σ x0) (x))),
(11)
3. Outline of the Proof Supercritical Case
where W is an N N Wigner matrix, (+)) E RV with (x(0, 0)) = dij. and 2 > ... > (8) > O. Let M be the transformed matrix defined by My- = N- N- J(NNM)) and let x = x (VNW(())1))))) Suppose that X
In this section, we outline the proof of the first part of our main result, Theorem 2.5. (The detailed proofs for the re- sults in Section 3 can be found in Appendix B.) Throughout Section 3, we assume A > 1.
1
(A) 0.
3.1. Approximation by a Spiked Random Matrix
It was proved in (Jung et al., 2024) that, almost surely, and (2) if
We begin the proof of Theorem 2.5 by approximating the transformed matrix M by a spiked random matrix. As dis- cussed in Introduction, it is required to approximate M by spiked Wigner-type matrix instead of spiked Wigner matrix, which is obtained from the Taylor expansion for the entrywise transformation f- The first step of this approxi- mation is the following proposition:
(1) i S E, then (M(M
i 1> f, then (M) + 2. (While Theorem in (Jung et al., 2024) is stated only for the optimal transformation of the form- -p'/p /p. the same proof works for any transformation.) The BBP transition for M would assert that (1) if i <
E, then 1/2 (M(
(V
1
converges in
Proposition 3.1. Suppose that M is a spiked Wigner matrix satisfying Assumption 2.3. Let M be matrix defined by M,j - J(VN[((N)) Define an N x N random matrix H by
VX
distribution centered Gaussian with variance 2(x 1)/A(") and (2) if i > f, then the fluctuation of (M) coincides with that of the (i (1-E) -)th largest eigenvalue of GOE matrix. However, avoid complication, we refrain from considering the rank-k models in this paper.
Hij = J(VNW,) + (VN(V (
VN
(12)
Remark 2.8. If Xe = 0 in Theorem 2.5, it is natural to con- sider a different scaling of the SNR for the BBP transition. Suppose that EVJ''vNWnn = 0 and (VNW12) # 0. With the scaling À = A(N) An /N, the Taylor expan- sion in (3) is then
Elf" (WWW
Let μi (M) and nn (H) be the largest eigenvalues of M and H, respectively. Then, for any > μ H1(M) (H O with overwhelming probability.
N S(VNN))
Proposition 3.1 basically asserts that in the Taylor expan- sion for f. the terms involving the third or higher derivatives can be ignored. Moreover, the terms involving the second derivative can only affect the largest eigenvalue via its ex- pectation. Note that it is an a priori estimate and we will eventually see that in the Taylor expansion, only the terms
J(VNW,))
1
2
This suggests that the effective SNR
X. = (/WW:)))
5

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Largest Eigenvalues of Transformed Spiked Wigner Matrices
involving the first derivative affect the largest eigenvalue, via its expectation, which justifies the approximation in (3).
3.2. Local Law
To estimate the change of the largest eigenvalue along the matrix flow given by Ht),, we use the Green function com- parison argument. It relies on the fact that the distribution function of the largest eigenvalue of a matrix can be well approximated by a certain functional of the resolvent of the matrix. (See Proposition 3.3 for an example of such a func- tional.) The main technical input necessary for the Green function comparison an estimate for the resolvent of the matrix, known as the local law. We will use the following local law for the resolvent of H (t).
Our strategy for the proof of Theorem 2.5 is to compare the largest eigenvalue of H with that of a rank-2 spiked Wigner matrix (in the sense of (11) for which the asymp- totic normality of the fluctuation of the outlier eigenvalue is known. The main obstacle is that the 'noise" matrix in H is not a Wigner matrix. To analyze this more in detail, let us introduce the short-handed notation
fi= = f(VNW,)) f===f ('VNWW) fi : f"
(13)
A/ - , (VNW.))
Proposition 3.2. Let (t) be the matrix defined in (17).
(From the assumption on the function f. we have that fl 0 for all i, and [[ff] = 1.) If we denote the noise matrix in H by V. whose entries are
Define the resolvent of H (t) by
G(t,2) = (1)))) =1)
(Im 2 2 0). (18)
fyy + VNSG - -Eff,Daa VN
Set = + in and
VO
(14)
K ki=T- C----VAELAA (VAELiA)) + VXEiSial
then the matrix V is a symmetric random matrix whose upper triangle entries are independent random variables (up to symmetric constraint), satisfying E[Vy = 0, but
Suppose that |x| S N -- and 17 = N 1 for fixed
1 π B[(V)))) = N
<> 0. Then,
(B( B[sy) B(D) mayy
VN
max (G) (0.00=maa(000)) = W
(19)
+ (B(()) -IfG)))))) 2?u}
1 N (1+ CY+ VNN,a, + Cÿ Nafaz).
(15)
uniformly on E [0, 1], where mac is the Stieltjes transform of the Wigner semicircle law, given by mse = (-2 2 + 4)/2.
Thus, V is not Wigner matrix but a Wigner-type matrix.
Although it might seem natural to interpolate between Vij and f /N, this approach is complicated by the corre- lation between fu//VI and the sub-leading order term vX(Si [= In this paper, we instead consider another interpolation, defined by a matrix ((t) for t E [0.1] whose entries are given by
For the proof of Proposition 3.2, we first prove corre- sponding result for the resolvent of V(() and then apply some resolvent identities. Define the resolvent of V (t) by
R(t, := (V(t) zf)-1 (Im 2 0).
(20)
A local law for R(t, 2) was proved in (Ajanki et al., 2017) where Riy 2) was approximated by a deterministic num- ber mt,t, z), given as the solution of the quadratic vector equation
1 + CV WN,,,, + CC 2
V(0)g= =
Vag. (16)
[V[(V
By definition, V(1) = V and (t) is a Wigner-type matrix for any / E [0, 1]. We can also see that V()) is a Wigner matrix whose off-diagonal entries are not identically dis- tributed. Note that it does not affect our proof since the fluctuation of the outlier eigenvalue is known for spiked Wigner matrices, even when the entries of the Wigner ma- trix are not identically distributed.
1 my ml(
Σ (())))))))m)(((
(21)
Note that if (V(())t) N- for all j. then (21) is sat- isfied by m,(0, = (=) for all i, since -1 11/mac (2).
For our purpose, the local law involving m, mt,t, 2) is not sufficient since we need an estimate on Ray (t, 2) that is uniform in j.j. To obtain such an estimate, we notice that while (1) is not a Wigner matrix for >> 0, it is asymp- totically a Wigner matrix in the sense that ((((((),,) 31 N-1 + O(N } }+2e). Thus, it is possible to bound the dif- ference (msc(2) - m, (t, 2)) and prove the desired local law for R(t, 2). (See Lemma B.3 for the precise statement.)
With (t). we finally introduce the desired interpolation, defined by matrix (t) for & E [0, 1] with entries
H(); V(0g + VVELAg
2
(17) Note that (0) is a rank-2 spiked Wigner matrix and H = (1).
6

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Largest Eigenvalues of Transformed Spiked Wigner Matrices
3.3. Green Function Comparison
for some constants Cm > O. Then, for any sufficiently small 0, there exists 8 > such that
With the local law in Proposition 3.2, we can compare func- tionals of the resolvents of (0) and (1). Before the comparison, first introduce well-known example of such a functional, in Proposition 3.3. We apply the follow- ing three-step procedure: (i) the cutoff of the eigenvalues by an indicator function on a small interval around the limit of the largest eigenvalue (by XE in Proposition 3.3), (ii) mollification of the cutoff function by the Poisson kernel (by 0g in Proposition 3.3), and (iii) approximation of the dis- tribution function of the largest eigenvalue by the (mollified) cutoff function (by K in Proposition 3.3).
E2 Tr Gt, x (el.riinnn in)dx
RF
Im
Bi
(22)
E2 + + in)
EF
Im
SN-
for any sufficiently large N.
3.4. Proof of Theorem 2.5 Supercritical Case
Proposition 3.3. Let (t) be the matrix defined in (17) and denote by pn (I) the largest eigenvalue of H(). Define
To finish the proof, we need to compute the eigenvalues of the spike in H(1). Let x² := (zî, 22..,....., Note that x and x² are not orthogonal, thus v [iz] may not be an eigenvalue of H(1) ((1).
.z2)
1 VIE[S12)=
1
L:= V No
+
Proposition 3.5. Lei AN == H(I) V(1)= V +
Fix c > 0. Let F E R such that |F 1F--I [.| N-1/2+e, Let
Al v VN= (x"(=")" Denote the ordered eigenvalues of AN by 0, 02 2 0N. Then, for any > 0.
E+ : L+2N-112HH and define XE: :- X|E,E. E-] Set 171
111/2-c/2 and 1/2- N- 3c, Let K E- + o0) be
smooth function satisfying
1. 01 vTe -((-1++))
1
| 1/3
K(x)
02 O(N
[2| > 2/3
3. 0g ON = 0.
which is monotone decreasing function on [0, oo). Define the Poisson kernel 0, for n > 0 by
Finally, invoking known results on the largest eigenvalues of rank-2 spiked Wigner matrices (e.g., (Renfrew & Soshnikov, 2013)), we can complete the proof of Theorem 2.5.
0g(y)= Tly² + 12)
7
Then, for any D > 0,
4. Outline of the Proof Subcritical Case
(,) (()) S E-m))N-D and
In this section, we outline the proof of the second part of our main result, Theorem 2.5. (The detailed proofs for the re- sults in Section 3 can be found in Appendix B.) Throughout Section 4, assume A < 1.
OT (H)) s E-
for any sufficiently large N.
4.1. Approximation by a Wigner-type Matrix
Heuristically, if pn (H)))) > F., then Tr(xE 00²) ) is nearly 1. By applying K, we change it to O, so that it can approximate the indicator function ((((H)))) < E). Note that we assume K is smooth to ensure that the Green function comparison theorem is applicable. The precise statement of the Green function comparison theorem is as follows.
The first step of the proof of Theorem 2.5 is the approxi- mation of the transformed matrix M by a spiked random matrix H in (12) for which we can directly use Proposition 1. In the second step, however, instead of interpolating H and spiked Wigner matrix H()) in (17), we directly compare H and Wigner-type matrix V in (14). We will prove the following result on the difference between the largest eigenvalues of H and V.
Proposition 3.4. Let e > O and set r = N - Let E1, F2g E satisfy
Proposition 4.1. pe and and (vVV) be the largest eigen- values of H and V. respectively. Then, 0 S (i(H) Mi (V) < N-3/4 with overwhelming probability.
E Ni (e=
Let F. R + R. be smooth function satisfying
max (x)) S Cm C
(m 0, 1,
prove Proposition 4.1, we compare the normalized trace
7

---

Largest Eigenvalues of Transformed Spiked Wigner Matrices
of the resolvents of H and V. Recall that
5.1. Spiked Non-Gaussian Wigner Matrix with Entrywise Transformation
Hij + +Viijaaa + > (lL
Jij
We first consider a transformed spiked Wigner matrix with non man-Gausiian noise. The (rescaled) noise entries VNWa (i S j) are independently drawn from the sum of Gaussian and Rademacher random variables, whose density p is given by a centered bimodal distribution with unit variance. (See Figure I(a) for the graph of p.) For the spike, we sample a random N-vector x so that VN 's are i.i.d. Rademacher random variables, independent from the noise. We apply the entrywise transformation
VN
foj
Vig-
-VAS - EEflDa
VN
Set
G(z) = 2(1,2) = (H - =1)
R(=)= = R(I,)) = (V -
f= f=
(24)
From the resolvent identity on R G. we can prove that (Tr(R(=)) G(=))| < Tr (rr R(2)] with high probability. We then prove Proposition 4.1 by showing that if jn (H) is not sufficiently close to pr (V), then Tr G(z) > 'Tr R(z)] for some 2 close to P1 (H) (but not to μr (V)). We remark that Proposition 4.1 not optimal, and the bound can be lowered; however, we do not pursue this paper.
where the Fisher information Fj, of h -p' /p is defined by
F = FI:=ENNVWiiP / (p((x))2 p(x) dx.
(See Figure 1(b) for the graph of f.) We remark that f in (24) is the optimal entrywise transformation in the sense that maximizes the effective SNR Xe, and it also satisfies Assumption 2.4.
4.2. Local Law and Green Function Comparison
From Proposition 4,1, we can immediately see that the fluc- tuation of μr (ff) is governed by that of μn (V), which is expected be of order N To prove that the fluctuation of M1 (V), we estimate the change of the largest eigenvalue along the matrix flow given by (tt), defined in (16). We use the Green function argument, which requires the local law, analogous to Proposition 3.2, for the resolvent of V (t). The precise statement is as follows:
1.0
0.8
Proposition 4.2. Let (t) be the matrix defined in (16). Recall that we denote the resolvent of (t) by R(t, 2). Set z = T + in and & == T --3 2. Suppose that [m|| S N- ftc and 17 = N. -e for a fixed > 0. Then,
0.
(a) The graph of the noise density P
max (t) - (2) - 1.
(23)
uniformly t E [0.
With the local law in Proposition 4.2, we can prove state- ment analogous to Proposition 3.4. By adapting the strategy explained in the beginning of Section 3.3, we can then con- clude that the the limiting distribution of the fluctuation of 41 (V) is equal to that of the largest eigenvalue a Wigner matrix (0), which is given by the Tracy-Witee Widom distribu- tion.
-10
(b) The graph of the entrywise transform f
Figure I. The noise density and entrywise transformation used in the numerical experiment with non-Gaussian bimodal noise.
5. Numerical Experiments
We set N = 1024 and generate 5,000 independent trans- formed spiked Wigner matrices described above. For a supercritical case, we set SNR À = 0.8 with X ~ 2.902. The histogram for the shifted, rescaled largest eigenvalue N1/2 (vTe A²) is shown in Figure 2(a). For
In this section, we perform numerical experiments compare the theoretical results from Theorem 2.5 with empirical results using specific models.
8

---

Largest Eigenvalues of Transformed Spiked Wigner Matrices
subcritical case, we set SNR À= 0.1 with Ae R 0.363. The histogram for the shifted, rescaled largest eigenvalue (2) - 2) is shown in Figure 2(b). For both cases, the empirical distributions closely match the theoretical results. See Appendix A. for more detail, including the precise formulas for the noise density p and the entrywise transfor- mation f.)
empirical distributions closely match the theoretical results.
See also Appendix A.2 for additional detail.
0.-
02
0.35
0.30
0.1
0.25
0.20
-2
4
0.15
(a) Supercritical case
0.36
0.30
9:09
0.25
(a) Supercritical case
0.20
0.18
0.10
0.08
0.00
0.15
(b) Subcritical case
0.10
Figure 3. The sampled distribution of ju for Gaussian noise with N = 1024, À = 2.5 (left) and À = 0.15 (right), after shifting and rescaling. The line in figure 3(a) plots the Gaussian distribution introduced in Theorem 2.5, and the line in figure 3(b) plots the GOE Tracy- distribution.
0.05
0.00
(b) Subcritical case
Figure The sampled distribution of μn (M) for non-Gaussian noise with N = 1024, x= = 0.8(left) and À = Oright,,, after shifting and rescaling. The lines in figure 2(a) plots the Gaussian distribution introduced in Theorem 2.5, and the line in figure 2(b) plots the GOE Tracy-Widool Widom distribution.
6. Conclusion and Future Works
In this paper, we proved that the largest eigenvalue of the transformed rank-one spiked Wigner matrix exhibits the BBP transition in terms of the fluctuation of the largest eigenvalue. The limiting law of the fluctuation is given by the Gaussian when the effective SNR is above a cer- tain threshold, and by the GOE Tracy-Widom below the threshold. We also proved the precise formulas for the limit- ing laws involving the effective SNR due to the entrywise transform, and strong concentration estimates for the largest eigenvalues, known as the rigidity. We conducted numerical simulations with several examples to compare the theoreti- cal results with the numerical ones.
5.2. Spiked Gaussian Wigner Matrix with Entrywise Transformation
We next consider a spiked Gaussian Wigner matrix, en- trywise transformed by a polynomial. We let each noise entry be i.i.d. Gaussian and apply the mapping (xx (a2 3z 1)/vIl entrywise. The spike is again sampled using i.i.d. Rademacher random variables.
We set N = 1024 and generate 5,000 independent trans- formed spiked Wigner matrices described above. For a supercritical case, we set SNR À = 2.5 with A R 1.294. The histogram for the shifted, rescaled largest eigenvalue / - (v + }) is shown in Figure 3(a). For a subcritical case, we set SNR À = 0.1 with X R 0.350. The histogram for the shifted, rescaled largest eigenvalue N2/3 (1z - 2) is shown in Figure 3(b). For both cases, the
A natural future research direction would be to prove the corresponding results for rectangular models, which gen- eralize the spiked Wishart matrices. We believe that our approach would work for the analysis on the largest singular values of transformed spiked rectangular matrices. We also hope to remove several technical assumptions on our model in the future.

---

Largest Eigenvalues of Transformed Spiked Wigner Matrices
Acknowledgements
matrices: Convergence and nonuniversality of the fluctu-
ations. The Annals of Probability, 37(1):1-47, 2009.
The authors would like to thank anonymous reviewers for their suggestions and feedback. This work was partially supported by National Research Foundation of Korea un- der grant number NRF-2019RIASA102832 1028332 and RS-2023- NR076695.
Chi, Y., Lu, Y. M., and Chen, Y. Nonconvex optimization meets low-rank matrix factorization: An overview. IEEE Transactions on Signal Processing, 67(20):5239 5269, 2019.
Chung, H. W. and Lee, J. O. Weak detection of signal in the spiked wigner model. International Conference on Machine Learning, pp. 1233-1241. PMLR, 2019.
Impact Statement
This paper presents work whose goal advance the field of Machine Learning. There are many potential societal consequences of our work, none which feel must be specifically highlighted here.
Chung, H. W. and Lee, J. O. Weak detection in the spiked wigner model. JEEE Transactions on Information Theory, 68(11):7477-7453, 2022.
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Largest Eigenvalues of Transformed Spiked Wigner Matrices
Renfrew, D. and Soshnikov, A. On finite rank deformations of Wigner matrices II: Delocalized perturbations. Ran- dom Matrices: Theory and Applications, 2(01):1250015, 2013.
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Largest Eigenvalues of Transformed Spiked Wigner Matrices
A. Details of the Numerical Experiments
In this appendix, we provide the details of the numerical experiments Section 5,
A.1. Spiked Non-Gaussian Wigner matrix
The density function p()) of each noise entry is
5 (e (e-2(1-2/V3)* + -
p(x)
(25)
2/2x
If denote by N and R a standard Gaussian random variable and Rademacher random variable, respectively, then p the density function of a random variable
N
R.
v5
V5
which represents the convolution of rescaled Gaussian density function and a rescaled Rademacher density function.
An optimal entrywise transformation (up to constant multiple) (xx) for the transformed PCA is h(x) == p((x)) model, the explicit formula for h(x) is
For our p(z)
5 (o v5
2/v3) +
(2m-+2/v5)+
v5
h(x)
(26)
2(m
(n+2\\5)
Note that h does not satisfy Assumption 2.4, since Fn == E| NW 12) | R 3.628, which is the Fisher information of the noise entry. We rescale & by dividing it by vhh, and thus we are led to apply
2.62503 (ar
v5 ) - 2/-- + (x+ v5 ) (= +2/v5)
f(x) = h(x)
~
(27)
VFn
2/V6)2
(o+22//5)
Note that [[[ff]] 1 and E[fí2] 1.905. We remark that for the transformed PCA the threshold for the strong detection is (E[ff2)) ~ 0.276.
A.2. Spiked Gaussian Wigner Matrix Recall that we let
x² + 3x 1
f(x)
V1I
It satisfies Assumption 2.4, since
m4
7m2
6m
(28)
11
11
11
11
where my denotes the e-th moment of the standard Gaussian distribution. (Note that m4 = 3, m2 = 1, and mj = 0.) In 3_ Rs 0.905, and the threshold for the transformed PCA (R[((2)) 1.222. We remark that the threshold cannot be decreased from 1 when the noise is Gaussian.
this setting, [ रiT
B. Proof of Main Results Supercritical case
In this appendix, we provide the detail of the proofs of the results for the supercritical case. Throughout Appendix B, assume that A 2 > 1.
In Appendices B and C, we will use the following notions, known as the stochastic domination in random matrix theory, which are useful when making precise estimates that hold with overwhelming probability up to small powers of N.
Definition B.1 (Stochastic domination). Let
ç = (c(0 NNNNuEEUOO)
<= (c(c NEN,UEUM)
13

---

Largest Eigenvalues of Transformed Spiked Wigner Matrices
be two families of random variables, where UNN) is possibly N-dependent parameter We say that & is stochastically dominated by (uniformly in u) if for all (small) and (large) D > 0
(w)) > NCc( 1 N-D
sup
kEtriM)
for any sufficiently large N 2 No(E.
We write < ( or = (()) if ş is stochastically dominated by Ç.
B.1. Proof of Theorem 2.6
We first prove Proposition 3.1.
Proof of Proposition 3.1. From Taylor's expansion,
J(VNM)) = F(VNW + vAN2,E,)
VN
VN
(29)
J(VNW,)
VF /F (WWW (F
VN
Recall the notation
Juj = D===I(V 1== S(VNW,, m= 18 (() (kEZ*),
(30)
defined in (13).
Define H by
fij - VMlaar + AVN
Hg
2 M3.
VN
Then, by definition (M - -M = 0 (1 N x x For given > 0, let Riy be the event
fly (max Si N
i.j
Note that By holds with overwhelming probability. Thus, the Frobenius norm (Hilbert-Schmidr norm) of (M H) is bounded by
M - - S
Σ 1J
JT Na N|
C(max (f (f Σ | O(N
in
In particular, ||M
- 11| 1).
We next consider (H - H). Note that
AVN
(H - (H-H)
(3/ - E[f
2
Thus, for any unit vector (vi, U2, ...,... .N) E RR..
A/N
(v. (H HH))
Σ
2
i
If we let F be matrix defined by F N ((G)), then it is constant multiple of Wigner matrix. Since the matrix norm of Wigner matrix bounded by constant with overwhelming probability, by considering the vector whose i-th entry is x Ue, we obtain that
Σ
AV S
=0
N N
O(N-1).
2
4
14

---

Largest Eigenvalues of Transformed Spiked Wigner Matrices
Since an arbitrary unit vector, find that || H| = ON
So far, we have shown that || - H|] = 1), Then, applying Weyl's inequality (or the min-max principle for largest eigenvalue), we conclude that f4 (M) (1 O(N-1).
Next, prove Theorem 2.6 for the supercritical case by adapting the strategy of the proof of Theorem 4.8 in (Perry et al., 2018).
Proof of Theorem 2.6 Supercritical case. From Proposition 3.1, we find it suffices to prove that
[ (VTe+ Vx
Let A and A be the matrices defined by
VA Ub BIfDD,
Aij
N
Then, for any unit vector y (yi,.... ,NN), letting ..NUI,
w.. Av) = VN Σ
A. (N( S VNJAI |
Since |zs] < N-1/2, we have that ||=|| < N-
1/2 y = N-1/2, Further, since A is constant multiple of a Wigner matrix,
we find that ||A|] < Thus, we have (y, Ay) - - 1/2, Since y was arbitrary, it shows that |||| - N-/22
We now consider (H - A), whose entries are
Jüj VAElsAaa BUJj)N VN
H Aig
Note that (H - A) is rank-2 spiked Wigner matrix. We want to invoke Theorem 2.7 in (Knowles & Yin, 2013), which can be rephrased as follows in our setting.
Proposition B.2 (Theorem 2.7 in (Knowles & Yin, 2013)). Let M = W + vxxx=" + V where W is Wigner matrix and æ = (#1,.... ..N): y (yI,..., ,NN) E R with (x, y) = 0. Assume further that O < A2 < 1 < Aı and 2 = 2N for any i. Then, for the largest eigenvalue P1 (1(M of M,
(M(M)) - (VAL+ ) -
Since the variance of the diagonal entries of (H - A) is not 2N but N-1, we introduce a diagonal matrix D with entries
D== (V2 1) fs
VN
From the first part of Definition 2.1 (the definition of the Wigner matrix), we can easily find that [ual < 1. Thus, ||D]] < N-1/2, Now, if let yn ( (H- A + D) be the largest eigenvalue of (H A + D), then from Proposition B.2, we find that
pI(1l - (V
NN-
Since
[w)( - (H D) < A D = ||||| + ||I - N 4
(31)
this concludes the proof Theorem 2.6 for the supercritical case.
15

---

Largest Eigenvalues of Transformed Spiked Wigner Matrices
B.2. Proof of the Local Law and the Green Function Comparison Theorem The proof of the local law, Proposition 3.2, relies on the following lemma, which is the local law for the resolvent of (t)..
Lemma B.3, Let (t) be the matrix defined in (16) and its resolvent R(t, z) in (20). Then,
max (ty (=c(=)0,yl 0 t.j
(32)
where we assume that the conditions in Proposition 3.2 hold = inp.
Proof. Recall the quadratic vector equation (21). In the regime considered in this lemma, we have an estimate for the resolvent R(t, 2) that
max (t, m,,t, 2)0,ر1 = O(N O #*t). 1.
(33)
(See Theorem 1.7 in (Ajanki et al., 2017), where the parameters are given as k(z) = (1) and p(=) = (n) from (1.17),
(1.23), and (4.5f) (Ajanki et al., 2017).) For the solution m, = (1, of (21), we can consider an ansatz
mz(t, 2) msc mx) (t,
with 9; = s;(t, 2) < Mac. We then have from (21) that
Σ Pmmm = =(mac + 81) Σ ((() + 81) + 85).
2m +
From (16), applying the identity sM + mag 0, we further have that
m + N (moc + +s) (1
(1 + CÝ t. NNNa + C²2 t2) (msc +s) = 0,
which yields
(a mgc)s; + Msc + 8
Σs; (mxc + 8i)
CY
Σ (mac + O
(34)
VN
Summing the left-hand side of (34) over the index and dividing it by N,
(z 2m)
1 N
Σ 86
Σ 8
2
ctt NVN
Σ (mac st)arg ) -(NN-++21).
(35)
N
Since |r:| - by naive power counting. the last term in the left-hand side of (35) is O(N- Thus, N Σ. 8; CONN--224 Plugging it into (34), since the last term in the left-hand side of (35) O(N -NN-)+ find that 8 (NN-++22) as well. With this bound on we then bound the last term in the left-hand side of (35) by
2
CY
Muc Σ
Σ 8:Ts = O(N O(N--+66)
xz +
NVN
where we used the assumption that Σ, O(N)). Thus. from (35), we find that N = and plugging again into (34), we conclude that 8; = (N+2))))
Combining the estimate 8; = O(N-1+2e) with the local law (33), we now obtain (32).
Remark B.4, The local law in (32) can be strengthened further; we will use the averaged local law
N
W. (Ra(t,2) (t, -ON
(36)
N
16

---

Largest Eigenvalues of Transformed Spiked Wigner Matrices
for any deterministic vector (wi, W2...., WN) with max, |wy| and the anisotropic local law
Σ w. (1,
Σ mi(l, 2 10r OONN-+22
(37)
for any deterministic unit vectors (W1, W2..... . WN) and (u1, U2 ..... UN). (See Theorem 1.7 and Theorem 1.13 of (Ajanki et al., 2017), respectively.) With the estimate 8; m, Ma (NN1++2 that we showed in the proof of Lemma B.3, we find that the local laws in (32), (36), and (37) remain valid even if we replace m, by Mac-
With the local laws for R(t, 2). we now prove the local law for G(t, 2), Proposition 3.2.
Proof of Proposition 3.2. We apply the resolvent identity
R(t, 2) G(t, R(t, (H) - (tt))
(38)
which can be easily checked by multiplying the identity by (H(t)- =T) from the right and by (V(!) =7) from the left.
Let 2² = az.. .2....,,, ,xx) Note that ||> || S (max; |2, x||| = O(N 22 for any d' > Then,
((t) - v(t) VAEls jax"
B Va
Set R = R(t,,), 2). G = G(t, 2), and let e, be the i-th standard basis vector. Then,
Raj Gij (e,, (R ( G)e,) VAE[fial(e, Ra) Ge,) + E (x²) (x),ee))
(39)
We want to show that the right-hand side of (39) is negligible. First, by applying the anisotropic local law in (37), we find
(e,, Rx) = ON 20), (e,, Ro²) =
(40)
To bound the second term in the right- side of (39), it suffices to use the trivial estimate |G]] S <n-1= = N which
can be readily checked from an inequality (f) - = S Im n where μi (t) (i = 1,2,... N) are the eigenvalues of (t). We then find that
Ra?) O(N 11=211 ONN 3t3r).
(41)
We now estimate (ar, Gez). From the resolvent identity (38).
(x, Rej) (a,, Ge,) VALifflla Rx) (x, Ge₁) + A BEfTEV (x, R Ges).
Solving it for (x, Gej), we get
(x. Ge,)
(z. Re,) Re,)
2 Elf BI N (z. Rx (x? (Ge))
(42)
VTE(fl (ax, Ra)
should be noted that for our choice of 2, from the anisotropic local law, (x, Ræ) is near - -1/(VAE[faa) To obtain an estimate on (az, Ge we consider the imaginary part of (æ, Ro). From the spectral decomposition,
(x(t)
R = (x(((11)=2
where vn (1) is the normalized eigenvector of V()) associated with the eigenvalue na (V()). Since V(() is a Wigner-type matrix with [tV V(O) = o(1), it is not hard to see that p& (V(t)) 2 -3 for all k = 1,2...., N with overwhelming probability. Thus,
IJ (tt)
Im(a, Rx) = Im
Σ
1(x, (t)) Im
(x(V(())- -
He (v())
k
k
Σ (ec (tt)) (H)(V(t)) - 7)2 + 1,²
7
2 (t + 4)2
(r + 4)2
17

---

Largest Eigenvalues of Transformed Spiked Wigner Matrices
with high probability, since (V))) forms eigenbasis. We now have that
+ [/2] Ra)] 2 V Rx)) 2 Cn)
(43)
for constant C' with high probability. Furthermore, from the anisotropic local law in (37), find that
(x, Rx²) = Σ maa +O(N1++2 macc Σ
2
m, macl + O(N 1-2)
= =ON + N 2tc N 2+2).
Thus, following the argument we used to derive (41), we obtain
BL V (a, Rx²) (a",)e O(N (tN
(44)
Plugging (43) and (44) into (42), together with (40), we get
ax, Ge,)
O(N2)).
Finally, going back to the resolvent identity (39), from (32), (40), and (41), we conclude that (19) holds. This completes the proof Proposition 3.2.
Next, we prove the Green function comparison theorem, Proposition 3.4.
Proof of Proposition 3.4. Fix x E [E1, Eg). For simplicity, set
f E
G = G(t, + in),
X Im
Tr
From our choice of x and and the assumption on F, we have from the local law for G, Proposition 3.2, that
X ONNON, (()
O(NC))
for some constant C. (Throughout the proof, we use C denote positive constants independent of N, whose value may change from line line.) Differentiating F(X) with respect
d EF E dt
dx
E2 dG
(X)
E
('
Im
Σ
dr dt
dt
(45)
Ea
dH
OG dx DH(t),k
E
(X)
Im
dt
E1
jsk
From the definition of Ht) in (17) and (16),
dH(())s
C
Vj
H(t),k=
dt
2(1 CY C2 C [VEI(())+]]] CV +
V(1)kt.
2(1 + X1+ + Cý IN
Applying the well-known formula for the derivative of the resolvent.
aG.b
Gay GarGib ii
(H(1),k
Gag
ifj=k'
18

---

Largest Eigenvalues of Transformed Spiked Wigner Matrices
find that
B2
V 2(1 + CÝ IVNaaa C V(t dr.
(F Im
(X) iisk
(46)
To prove that the derivative of [F)(()] is small, Stein's method to bound the right-hand side of (46). Notice that
dG.b 8Gob
@V 3H(1),²
which can be readily checked from the definition of (() in (17). Let
Ω {max Gij myy < N 7 for all [E1, E2]}.
Then, Qe holds with high probability. Ωe when we expand the right-hand side of (46), the terms involving the third or higher cumulants are negligible in the sense that is O(N O(N-$+C0) Indeed, if we perform the naive power counting, we get a factor N from the third (or higher) cumulant, & from the length |Ez - Eil of the interval in the integral, N from the three summation indices j, k, - +2r from the factor (CY VNzzz + Cý Nx}æz). and N-I+Cc since each term contains least two off-diagonal entries of G, hence the total size of these terms at most (While some terms may contain less than two off-diagonal entries in case some summation indices coincide, the power from the summation indices decreases to N in such cases.)
- N - N-I-C() = ONN
From the power counting argument above, on Ωe, we find that
d EF(X)
dt
En {(((()))))
+ CP +
=-Im
Im
E 2(1+ + + CVN.NN C. (N)3333)
(F(()) dx (V(1))k
Ei
f.J.k
OON
B2 E(CY + C² N2f=1)E FX)
0x OV (t))),
Im
GiyGRi + F')
dz
2N
(t)))
E1
+ OON
(In the rest of the proof, we always assume that Ω2 holds.) We notice that,
ax (v(g)
F [GG + Gar = O(N D(N-++0)))
Im
from the power counting; here the factors are - from the length [Eg - Eil. N from the summation index, and N --1-Ce from two off-diagonal entries of G. (The notation above illustrates the case where j & k;; however, we easily see that the same calculation applies for j = k.)) Then, again by power counting.
F2 1 N
1
0x OV());
+ Cy Nafof)B [x(
GiyGrs dr O(N-
F1 ij.k
remains to bound
1
Es
Σ CCY VNNazk CC NaJaRiE F()) ((G,)Gas))
dx
2N
@v
(47)
VNNa (N G.G dr.
E
19

---

Largest Eigenvalues of Transformed Spiked Wigner Matrices
For the term with three off-diagonal entries of G, we can again the power counting show that is O(N- Similarly, with the factor Na ONN-++1e) find that
1
Fi
ij.
For the remaining term,
Σ [VV
N F
1
(48)
N
E1
Σ Li.k
B - -mae
N
The second term in the right-hand side of (48) is ONN +Cee again by power counting, where we use the local law macl O(N-2++C),, proved in Proposition 3.2. The first term in the right-hand side of (48) can be bounded as
|Gya=
E
Σ
dr
N
Ey
$.J.K
CV
Es
Mys
Σ
Σ i
EG (X O(N O(N-2009
VN
E1
where we use the assumption that Σx Iz O(N)).
So far, we have seen that di EF = 1ON--++C for any sufficiently small > Integrating it from = to 1, we can conclude that (22) holds. This completes the proof of Proposition 3.4.
B.3. Proof the Main Theorem Supercritical case
We begin by proving Proposition 3.5.
Proof of Proposition 3.5. It is obvious that 0g ON since Ay is a rank-2 matrix. To ease the notation, we assume that E[ 2 and denote An un." voT, where
u (VAB((2)) x
(49)
v 2 (ELTTVVN)
(The case E.f% < 0 can be proved in similar manner.) It can be readily shown that the nontrivial eigenvalues of An given by
| + ||a||2
2
1
23 + 4(2, w)2
By substituting u and v, we get
0, vYElSi] AE
24)
AE ((N,
(50)
and
8 VXE[F12] +
VN x/))
/ AE P
O(N Φ
(51)
20

---

Largest Eigenvalues of Transformed Spiked Wigner Matrices
where we define
X2 E 1 2 N xx,) (+)
+AAAE1/511EEAAAAA Σ 23/2- Σ 1!))
Note that for any > O,
((N, - -2(VAL(f1))
S at)) 11
(52)
Then,
(1+ (1+ @(N,z) + +O(N-I+t)
V + ViBlfial
2(v
(53)
= -ABifial - - Σ 2t) + +0(N
and plugging it into (50), get
0, -
(54)
and
02 7 B 2
V =
Σ
(55)
This proves the desired proposition.
The proof of Proposition 3.3 essentially follows the same strategy as the proof of Proposition 7.1 in (Lee & Schnelli, 2018). However, since the asymptotic spectrum is determined thanks to Theorem 2.1 of (Capitaine et al., 2009), we take a more straightforward approach as follows.
Proof of Proposition 3.3. Let (t) = (t) be the k-th largest eigenvalues of (t).. Recall that in the proof of Theorem 2.6 in Appendix B. we have seen that | - (H + D)Il < N-1/2 and (H A + D) is a rank-2 spiked Wigner matrix. By Proposition 3.5, Theorem 2.1 in (Capitaine et al., 2009), and Theorem 2.7 (Knowles Yin, 2013), with overwhelming probability,
(px(1)= - Ll > L L-2
2
for k = 2,.,...., N. To prove the proposition, compare (XE (Ni) and XE-m for the following cases.
Case I. For < L+2 XE-m (x) = O. Then we get
(XE * (x))
1
lan -1
ES tan
E - x
1
(an
1/2
Lan
1/2
1/2
1/2
E.
E4
(56)
1
1 me (2(EE
E)
N <N-133 3
2
E
E4 3
2 2 x)(EE, x)
Therefore,
Σ
(XX (Ai) - XE-m (di)) S N 2N-1- 3 N
(57)
E: LARKE= L-2
Observe that only one eigenvalue of Ht),, namely f1(t), be in the following two with overwhelming probability. Case 2. For E 2 E m)), XE-m (x) = O. Similarly the formula shown above,
(XE * (x))
1
lan -1 E+ -
tan -1 E E-sx x 1
(an -1
7/2 tan
1/2
1/2
1/2
E
E.
(58)
1
1 m2 (2 - E)
S N
2
E
E1
2 (E
x) EE_- x))
21

---

Largest Eigenvalues of Transformed Spiked Wigner Matrices
Therefore,
Σ
(Xx * (2a) XE-m (Ai) S N
(59)
Case 3. For 2 E |E 1,, E4), we use the trivial estimate
(XE (z) < Xe-m (z).
(60)
Case There are no eigenvalues in [E+, 00) with overwhelming probability. Considering all cases above, with overwhelming probability, we observe that
Tr(x= * (H) & Tr Xx- (H) + N
(61)
Since Tr XE-m (H) is an integer, by the definition of the cutoff K,
K K XE-m (H) + 9 =
where M(E-m.E) = (i: d E EE- f1, E+]] Since is monotone decreasing on [0, oc), (61) implies that
K(Tr(Xe 02)(H)) 2 K(N(E-m.E.)
with overwhelming probability. By taking expectation both sides, find that
[[K(Tr(X (2)) P()1 E --m)- -
for any D > The second part the proposition can also proved using similar approach by showing that
Tr(xe (HH) 2 Tr X (H) - N-
To finish the proof of Theorem 2.5, we use the following result on the fluctuation the outlier eigenvalues of spiked Wigner matrices. Let
82
1
CO: 0
Pj 0, Aj
Proposition B.5 (Theorem 1.3 in (Renfrew & Soshnikov, 2013)). A .... An be ordered eigenvalues of W + AN-
Let u'....,,u, uN be a set of orthogonal eigenvectors of AN with eigenvalue 0, > 1. When the eigenvectors of AN are delocalized, the difference between the k dimensiiil vector
(co, VN +. pj), i == 1,....,,k
and the vector formed by the (ordered) eigenvalues of ki x ki GOE matrix with the variance of the matrix entries given by - plus deterministic matrix with ipth entry (1 1 L, S k2) given by T. (u'y) Mauyy converged to zero in probability
8²
with
(Ma) =
-
We now prove the first part of Theorem 2.5.
Proof of Theorem 2.5 Supercritical case. Fix > and let ? = N-1/2-e/2 and 7/2 = 1-1/2-3e Consider L + NN-1 with (-Ns, and E+ L 112+e, Recall that px(() nh (t) is the k-th largest eigenvalues of (t)). By Proposition 3.3, we get
P(M((1) < F.) EK(TT(XEtm (H)) N-D
22

---

Largest Eigenvalues of Transformed Spiked Wigner Matrices
for any D 0. By applying Proposition 3.4 with 3e instead of e, find that
N
E+ TrG inz) dæ
(K(Tr(XE+m xH)) EK
Im
E+m
(62)
N
/ TrG(O,
SEK
Im
im2) dr
+ N EK (Tr(xEtm 1Tp (H())) + N
E+m
for some O. We can apply Proposition 3.3 again (0). Consequently, obtain
IP ((N1/(((1()) - L) (1) -1 - E)
EK(Tr(XE+n (H())) +N < EK (Tr(XE+n *0,) + - + N
(63)
< (0) = + N (N - L) L) +2N +)) N
Similarly, we also find that
TP (11 - L. 5>)
-0)
L. - 2N N
(64)
Since the fluctuation of uu (0) coincides with that of μ1 (M) from Proposition 3.1, remains to prove the limiting distribution of (H))) We apply Proposition B.5 to the largest eigenvalue μ1 (0) of (0)) V(0) + Ax to find that the fluctuation of
N \ (0) -
coincides with that of a Gaussian random variable with variance / 1). Further, from Proposition 3.5, we can replace 9, by v with negligible error. To find the mean of the limiting Gaussian, we need to compute the deterministic term involving M3 in Proposition B.5. For simplicity, let
3s = (0))
so (Ma)i) = -1a(1 5.y) since V(0) is a real Wigner matrix. Then,
N
T
M3
82N
73
=
-
(65)
with the assumption Σ, O(N5). We thus find that the mean of the limiting Gaussian is 0, which concludes the proof of Theorem 2.5.
C. Proof of Main Results Suberitical Case
In this appendix, we provide the detail of the proofs of the results for the subcritical case. Throughout this appendix we assume that A₂- >(E[(f())
C.1. Proof of the Local Law and Theorem 2.6
We begin by proving the local law, Proposition 4.2.
Proof of Proposition 4.2. Recall that we have seen in the proof of B.3 that [m;(2) msc(2)] x O(N-1), Since this estimate does not depend on 2,
max R.y (t, 2),,,1 = O(N $t)),
which was to be proved. (See Theorem 1.7 in (Ajanki et al., 2017), where the parameters are given as k(z) = O(1) and p(z) ((V + 7)) from (1.17), (1.23), and (4.5d) in (Ajanki al., 2017).)
23

---

Largest Eigenvalues of Transformed Spiked Wigner Matrices
We first notice that Proposition 3.1 holds for the suberitical case without any change. Thus, Theorem 2.6 is simple corollary of Proposition 4.1, since from the rigidity of the eigenvalues of Wigner-type matrices, Ian (V) N-2/3 (We refer to Corollary 1.11 in (Ajanki et al., 2017) for more detail.) We now prove Proposition 4.1.
Proof of Proposition 4.1. The first inequality of the proposition, μ1 (HI > (i(V), is obvious from the min-max principle for the largest eigenvalue. We now suppose that μl (H) (1 > N N-3/4.
Fix > O. We choose the parameter = = 7 + in with r = fai (V) + & for 5 E [N-3/1 NN1/2++ and = N-5/6 Set R = R(=) = (V 21) G = G(z) = (H - =f) Our first goal is to show that R - Gul = O( 1/3). To prove
this, we start with the resolvent identity
X BlflalNN (x)) Ge,), R3 - Ga = (e;, - G)e)) = VAElfi2 Rx) (a. Ge,)
(66)
2
which is obtained by letting = j in (39). To estimate the terms involving R. in (66), we apply the anisotropic local law, Theorem 1.13 in (Ajanki et al., 2017), which is
kw, Ro) - mea(2)(w, (w, w)1 < N-1/
(67)
for any unit vectors w, v. (Here, the parameters are given as x(z) ((1) and p(z) O(1) O(N 5/6) from (1.17). (1.23), and (4.5d) in (Ajanki et al., 2017).) From (67), we find that
(a, Re:) = O(N-110) (x, Rx²) (NN2
To prove an estimate for (ar, Gej), we use bootstrap argument. (42) in the proof of Proposition 3.2, by using the resolvent identity in (38), we had
(Ir, Rej) A VESS)IN((2. (x?) Ge.))
(x. Ges)
(68)
1+ VXE[fi2l(x, Rx)
From (67), (x, Rx) -1 o(1) with overwhelming probability. Since
vAE[f12] we find that
1 + (AEl/fi2( Ra)| c>
with overwhelming probability, for some (N (N-independent) constant Then, applying the trivial bound ||G]]| 1n 1n-1=10 N5/6 from (68), we find (x., Ge:) ON1/9), which serves as an a priori estimate. To improve this bound, we consider
(a² Re:) - (x Ge,)
using the resolvent identity in (38) and solve it for (x: Ge.). Then, get
Ye2,Ge 1 EE (x22 Rw.2)
1
(x², Re,) VAE[Sial(?? Rx) Ge:)
(69)
We notice that (x², Rx?) O(N- 1) 2x², Re,) = 2/3), and (x,,Rx) O(N 2/3) from (67). Thus, from the
priori estimate (x. Ge,) = (NV1/)), we obtain that (x? Ge.) = O(N-1/2) Plugging it back to (68), we now have that (x, Ge.) = O(N-1/6)
So far, we have seen that
(z, Ge,) O(N-1/6), (a², Ge,) -NN2
Plugging these back into (66), we finally obtain that RR Gal = ON/33) In particular,
T Tr GrG = (NN233).
(70)
To prove the desired theorem, consider
N Σ xm1
Im Tr R
(H,(V) 7)2 + n?
24

---

Largest Eigenvalues of Transformed Spiked Wigner Matrices
where P1 (V) 2 2 uN (V) are the eigenvalues of V. We now find an upper bound for Im Tr R by partitioning the
eigenvalues (p:(V)) into the following four groups:
E (V)(V) : 0 S ()()V) - -i(V) S N N
Es = (v.(V): N < - S SN-5/9)
33 = (V) N (V) - (v) 2/9), Es = (Mi(V) : (V) Mi(V > 2
From the rigidity of eigenvalues (V),,, we have that
E = O1), Egl O(N1/6), [Es]] = O(N1/), | O(N). 5/6
Recall that we are assuming 77 N 5/6 and pM (V) = = N-34 > 7. Then, for any eigenvalues in Et.
(ui(V) 7)² + 12² 2 3/3
3/2 +1² > 3-/2
For the eigenvalues in E2, and similar estimates hold for the eigenvalues in Ez and E4, respectively. Thus, we get
(v: 7)² + n 2 (V:(V) μu N 1/3
Im Tr
Σ+Σ+Σ++
7 (v:(V) - 7)
E1
E2
Es
E1
(71)
= ON
8/N 3) + O(N N N/N $) + O( + N 1 NON'N-B/NN
Now, suppose that (r [T, + n/. Then, with in.
Im G Σ
11 (m((H)-]
17
1
N
1/ (H) - -r)² np
21
2
which does not hold with overwhelming probability, since Im Tr = O(N2/3) from (70) and (71). Considering O(N 13tc) such intervals, we can conclude that
(H & (V) + 3/4 3/4 (V) + N-1/2t
with overwhelming probability.
Finally, adapting the strategy of the proof of Theorem 2.6 in the supercritical case, we can prove that (H - 141 [1(() < 1/2, N-1/2, This completes the proof of the desired theorem.
C.2. Proof of the Main Theorem Subcritical Case
We begin by introducing a result analogous to Proposition 3.3.
Proposition C.1. Let (t) be the matrix defined in (16) and denote by p1 (t) the largest eigenvalue of Vt).. Fix e >
Let E R. such that E 2| < N -2/33e,, Let E+ ;= 2 + 2N -2/3+e and define XE = X|E,E+) Set m := N 2/3--/22 and n2 = N 2//33-3, 3r, Let K and B, be defined as in Proposition 3.3. Then, for any D > 0,
E[K(T 0z) ((p((((()) S E - - - N-D
and
[T * og (H))) < (H < - -N
for any sufficiently large N.
Proof. The proof is standard; see, e.g., Proposition 7.1 in (Lee & Schnelli, 2018), with straightforward adjustments by applying the results in (Ajanki et al., 2017).
25

---

(F'( Rkd) dx Largest Eigenvalues of Transformed Spiked Wigner Matrices
Next, we introduce and prove the Green function comparison theorem, analogous Proposition 3.4. Proposition C.2. Let and set 1 - Let E1, E2 E R satisfy
[E 12-22 2 S + 1,
Let F:; P:R IR. be smooth function satisfying
max (F(m) (a)) S Cm (xl)
(m = 0,1,2,3.4)
for some constants Om > Then, for any sufficiently small e > 0, there exists > O such that
Es Tr R(1, in)ddc
Bs
EF
Im
EF
Im
Tr R(0, x in)dar
S N-6
(72)
E
The proof of C.2 almost a verbatim copy the proof of 3.4. We nevertheless present here a detailed proof of C.2 for completeness.
Proof of Proposition C.2. Fix E2]. E2). For simplicity,
= Ri(t a in),
Im
Tr R(t, + in)
F
Then, x= = O(NC), (X) = O(NC)) for some constant C. (Throughout the proof, we use C' to denote positive constants independent of N, whose value may change from line to line.) Differentiating F(X) with respect to t, we find the following formula, anaogous to (46):
E2 E
CY + Cy V(t) dr.
dt EF(X) = BF(X)= Im
F'XX
(73)
2(1 + CY I WNNN + C
kjjh
Applying Stein's method on the event
Ωr = {mmx R.
meddizl < N
+ for all x E [F1, E2]h
1.
we expand the right-hand side of (73). In the expansion, the terms involving the third or higher cumulants are negligible in the sense that is ONN- which can be proved by performing the naive power counting. (Alternatively, we can rely on the edge universality for Wigner-type matrices to assume that the entries of V are Gaussian for which third or higher cumulants vanish; see Theorem 1.16 (Ajanki al., 2017) and the remark below it.)
From the power counting argument as in the proof of 3.4, on Q, we find that
d BF(X)
di
+ CC + 2(1 2(1+C) Cy (Y
a av(1),
=-
B((V(L)))
E1 i.
+ONN S+CC
Es [CC Cy Nefat)E (x)
0X (V())
Ray F'(X) 0 0(R dx @v
2N Im
fjj.
+O +O(N-3+C9)
(In the rest of the proof, we always assume that Ω2 holds.) We notice that
0x
Eg Σ
---Iin Im
Rag Rum + Run Ryo) )dx 0(N-)+0)
(V())k
E
26

---

Largest Eigenvalues of Transformed Spiked Wigner Matrices
from the power counting, and also
1 Fi N F.
ax (V)))
VCNJaa + O. NaJaR)E (X)
Ry Rki Rxi
dx ((o)).
iisk
For the main term, which coincides with (47), we apply the same expansion as the proof of 3.4. Then, we can find that
E2
Σ C
- -ON (F
N
For the remaining term,
CCY
Cy CY VNryaRRBB (X) Rax Ryd dr 0(N
N
1.s.k
by splitting Rak into mis and (REs- msc) as in the proof of 3.4.
From the estimates above, we find that (EF(X)= = -(N-3+Ce) for any sufficiently small e > 0. Integrating it from / to we can complete the proof of Proposition C.2.
at
Finally, we prove the second part of Theorem 2.5.
Proof of Theorem 2.5 Subcritical case. Proceeding as in the proof of Theorem 2.5 in Appendix B.3, we can prove that the limiting distribution of pa (M) coincides with that of (4 (V). Since is a Wigner matrix, it is well-known that the fluctuation of jn (M) is given by the GOE Tracy-Widom distribution. (See, e.g., Theorem 1.2 in (Lee & Yin, 2014).) This proves the desired theorem.
D. Discussion on Different Scaling for the Effective SNR
In this Appendix, we analyze the case A and À \oVN, considered in Remark 2.8, more in detail. Recall the short-handed notation in (13) and the assumption that E[fiz] 0, E E o For simplicity, further assume the i.i.d. prior, where vNx; are i..d with mean 0 and variance 1, and all moments of /Na, are finite. While discuss mainly the supercritical case in this appendix, the subcritical case can also be handled in a similar manner.
D.1. Approximation by Spiked Random Matrix
From Taylor's expansion,
N/ 1)/2
Mi =
e!
li
(6) (0) + O(N1
t=
fy
1/2 (3e- (38-2)/2 23/4
X
(36 (36-2)/4 2)/4
(74)
-
-
10
Dafaf
B 14fff + +(N1331773)
VN
!
A
t=1
2=
=: Hi + 13/13/112220
Note that H in the right-side of (74) is different from the matrix H defined in 3.1, but the notation only in this appendix to make the comparison between the case X 0 and the case X + O. As in Appendix B.3, the Frobenius norm of (M- H) is bounded by
| 11| =0 N113/4
Σ
r x
10)
1/2
((N3314,
which will be negligible when compared to the fluctuation of the largest eigenvalue. This in particular shows that H an approximation M.
27

---

Largest Eigenvalues of Transformed Spiked Wigner Matrices
As next step, we denote the noise matrix in H by V, whose entries are
J
2/2 N(3--2))
Vij
U
E
Again, we use the as the equation above only this appendix to ease the notation when comparing the cases d. = and d + O. The matrix V is a Wigner-type matrix with the variance
1 = N C N may + C. 2 CY CY +Y O(N-9/4).
Here, we notice that the variance of Viy in this case is significantly larger than that in the case de # 0, computed in (15). Indeed, if we compute the sum of B[((,;))) we find
Σ J.
=
N C (NV++01))
Note that
C = B[Siaff B(fia-
(75)
Since the sum of the variances of the entries of Wigner matrix is N + O(1), it means that the limiting empirical distribution of V is not exactly given by the semicirele law but is stretched by the factor of (1 + C. N-1/22-1//2 To compensate this factor, introduce the interpolation matrix defined by
v(t) = (1 Ch 1N1
11/2
1+
4
CYENS//LOfaf
Vy
[V,[(V
t=1
Note that
(((V()))) N + CKtN 1/4 n,, + CYtVN
[71) N2
From the definition V (t), we have
= 11+ cx 1111 1/2 ((1).
(76)
D.2. Local Laws
We now continue prove the local laws for V(I), the estimates corresponding the statements in Lemma B.3 and Remark
B.4. Following the proof of Lemma B.3 in Appendix B.2, with the ansatz m, 2) mec(2) + st(t, we find that
CYt
+ m ms
Mc Si
ΣB + (mac + 81)
85
(msc + 83)
N
N1/4
+ (moc + Ci) tVN Σ (mec + (z)(2
1
N2
(77)
(z msc)s +
Mse + 34 Σ 89 (mic + si) N1/4
Cyt t
Σ
(mze + 5;)
+ (mye + C tv (
)) + (m + 8: ) CCCty
N Σ (2f2n N22 =
O(N-1).
which corresponds to the estimate in (34). Summing the left side of (77) over the index and dividing it by N, we find that
2
cY
(z + 2maaa
N
Σ
8:
+
N
8;
+
N5/4
Σ (mac + 81)
(78)
2m 2y
Σ
s:(a
CXt
8183 (n7
==)= = O(N-1), =
1
VN
VN
28

---

Largest Eigenvalues of Transformed Spiked Wigner Matrices
which gives 2 E, 8 - ((N by naive power counting. Plugging it back into (77), we also find that 8;
(NN-1/4
as well. (Note that these estimates are weaker than the ones had in Appendix B.2.) By following the bootstrapping argument in Appendix B.2, we can improve the estimates to N Ee 8; bootstrapping again, we find /V Σ; 8 O(N- and s; = (N 3/ Now, combining the estimate s; ONN 3/4) with (33), we conclude that the entrywise local law (32) and the anisotropic local law (37) hold for the current case.
O(N-3/4) and 8; O(N ((N). Finally, by
With the entrywise local law and the anisotropic local law for R. the resolvent of V. we can also prove the local law for G. the resolvent of H, as in the proof of Proposition 3.2 in Appendix B.2. Here, we have
H()) v(0)) + (1+ C. 1/2)
-1/2
No 3
N//NT/4
EL 13]3 +
1/.N1522
E
14) Ju]of]
and in particular, (0) is rank-3 spiked Wigner matrix and
1/2 H H (1 Cy CYN-112 H(1).
D.3. Green Function Comparison
With the local laws for R, we can prove a statement analogous to Propositions 3.3 and 3.4. Indeed, the proof of Proposition 3.3 does not change in the current case. To prove Proposition 3.4, we can proceed as in Appendix B.2 and find that the most part of the proof remain unchanged, except that the error bound - ++Ce changes to O(N O(N-++Ce) The only nontrivial term that should be considered separately is
1
E ENN 1 (Hii
N
which is O(NCe) from the naive power counting. To obtain a better estimate for this term, we adapt the idea in (48) to find
Σ
N
1 (x) dx
N
E
kjjkk
1
Σ
No NE LE [F" (X) Gydld
N
i.jkk
1
Σ J.K.
C N
1 N2
B F - mec) dr.
N
E1
The second term in the right side of the equation above is O( ++C) by the power counting, where we use the local law for Gka mss- The first term in the right side is further decomposed into
1
C
N
NO
B
Σ No N EF
(X 1 E
Ecy - 1 [F()
mayma da
JE
N JE
E
1
E2 1e
Σ
C.. N(.
N (F -m/e) dr + N NJE, Σ NON E mec dr.
Note that the second term in the right side of the equation above vanishes due to the summation of (1² over the index To estimate the first term in the right side, we notice that (²(= Gz).. Thus, by applying the Cauchy integral formula for the derivative of G' and mae along the square contour centered at z with the sidelength 7), together with a lattice argument on the contour, we can prove that (G²) m'r = O (The proof of the local law for the derivative, based on the Cauchy integral formula and the lattice argument is standard in random matrix theory; see, e.g., Appendix B of (Chung & Lee, 2022).)
29

---

Largest Eigenvalues of Transformed Spiked Wigner Matrices
D.4. Fluctuation of the Largest Eigenvalue
Recall that the largest spike of (t) is (1+ (LL00()(((NN))2))/)))))))((" If we let B[ag]] W4, the matrix norm of the largest spike of (t) converges to A
Following the discussion in the previous appendices, we can conclude that the fluctuation of in (H())) coincides with that of u1 (0)), and thus it satisfies
/ (1(H(L)) - (V + -
+ +N(0000
with 0² =2(xe -1)/ee, where the effective SNR Àe is given by
X =
As the last step, we need to multiply the statement above by the factor (1 + C2 N-1/2)1/2, introduced in the definition of V (t); see (76). can thus conclude that
(M(M) - (Vio+ +
N2/2
V
N(
(79)
where 0² = =(Ae - 1)/Xe and the mean of the limiting distribution is given by
(V de +
2
X
The argument used in Appendix also applies the subcritical case, which shows that the largest eigenvalue converges to 2 and its fluctuation converges the GOE Tracy-Widowe distribution after it is shifted by Cý² 1-1/2, Note that the deterministic shift Cý 1-1/2 is much larger than the size of the Tracy-Wo Tracy-Widom Widom fluctuation, which is of order 2/2/3
Finally, we remark that the argument in Appendix D can be naturally adapted to a more general case. we let
ky = in E zt
0},
then by following the argument in Appendix D, it can be readily checked that the transition happens in the scaling (Note that we had assumed ky = 2.) Moreover, we can also prove that the BBP transition in the level of the fluctuation also can be proved in a similar manner, though the proof would require more involved analysis with more levels of bootstraps. We also remark that an appropriate 'correction factor', which was (1 + Cý N-1/2)1/2 for the case f = 2, should be applied to find the deterministic shift for both the supercritical and the subcritical cases.
= No.V 2 (1-
30


Reviews:
[{"review_id": "LpFyktV4od", "reviewer": "Reviewer_dwXH", "summary": "This paper studies the fluctuations of the largest eigenvalues in transformed spiked Wigner matrices and discuss the Baik–Ben Arous–Péché (BBP)-type phase transition arisiring in these problem. While a great deal is known about these matrix, the contribution is the analysis of the asymptotic fluctuation  of the leading eigenvalue, proving that it follows a Gaussian distribution in the supercritical regime (when the effective signal-to-noise ratio is above a critical threshold) and the Tracy–Widom GOE distribution in the subcritical regime.Their approach relies on precise concentration estimates, interpolation arguments, and Green function techniques to track the eigenvalue fluctuations.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for the constructive review and helpful feedback with which we can greatly improve our paper. Below, we address the concerns and questions, and also outline some important revisions we will make.\n\n**Relevance to the ICML audience**: Thank you for the comment. In the revision, we will incorporate the paragraphs below into the introduction of the current manuscript to demonstrate the relevance of our model to the ICML audience.\n\nSpiked random matrix models play a crucial role in analyzing for many machine learning problems. For instance, in the study of the feature learning in a two-layer neural network, a rank-$1$ perturbation of the initial weight matrix can approximate the updated weight after one gradient step (Ba et al. 2022). This idea is further developed by (Cui et al. 2024; Dandi et. al., 2024), where the spiked Random Features model was considered with first layer weights of the form $X + r \\boldsymbol{u}\\boldsymbol{v}^T$. Additionally, these models have practical applications in machine learning algorithms, such as an ensemble method for Q-learning in reinforcement learning (Lee et al, 2023). \n\nThe spectral properties of spiked random matrix models are important in analyzing various problems in machine learning and statistics, as spectral methods can improve convergence analysis and provides theoretical guarantees. Notable examples include the community detection (Abbe, 2017) and submatrix localization (Butucea et al., 2013). In the context of machine, we refer to (Mondelli and Venkataramanan, 2021) for an application to the approximate message passing algorithm and (Chi et al., 2019) to non-convex optimization.\n\nEntrywise transformed random matrix models provide valuable insights into the study of fundamental limits of detection for the signal from a spiked random matrix, which has been extensively studied by analyzing the eigenvalues (Montanari et al., 2015; Johnstone and Onatski, 2015) or the mutual information and minimum mean squared error (Lesieur et al., 2015; Krzakala et al., 2016; Dia et al., 2016; Lelarge and Miolane, 2017). Further, these models contribute to the theoretical analysis of deep neural network, where entrywise transformations correspond to pointwise nonlinear activation functions (Pennington and Worah, 2017).\n\n**References**: Thank you for the reference of (Mergny et al., 2024) that we missed. We will include it in the revision.\n\n**Question on generalization**: Thank you for the good question. It is also possible to prove similar results with different SNR scaling. In our setting, the index $k_f$ can be defined as $k_f := \\inf \\\\{ k \\in \\mathbb{Z}^+ : \\mathbb{E}[f_{12}^{(k)}] \\neq 0 \\\\}$. Under a mild assumption on $\\boldsymbol x$ (e.g., $\\sqrt{N} x_i$'s are i.i.d. with an $N$-independent distribution whose mean is $0$, variance is $1$, and all moments are finite), the critical scaling of the SNR is $\\lambda \\equiv \\lambda(N) = \\lambda_0 N^{1-1/k_f}$ and the entries of the spiked Wigner matrix are given by $M_{ij} = W_{ij} + \\sqrt{\\lambda_0} N^{\\frac{1}{2}(1-\\frac{1}{k_f})} x_i x_j$. Let us consider the case $k_f = 2$ for simplicity. The Taylor expansion in (3) is then $$N^{-\\frac{1}{2}} f(\\sqrt{N}M_{ij}) \\approx \\frac{f_{ij}}{\\sqrt N} + \\sqrt{\\lambda_0} N^{1/4} f_{ij}' x_i x_j + \\sqrt{\\lambda_0} N (f_{ij}''- \\mathbb{E}[f_{ij}'']) x_i^2 x_j^2 + \\sqrt{\\lambda_0} N \\mathbb{E}[f_{ij}''] x_i^2 x_j^2.$$ This suggests that the effective SNR $\\lambda_e = \\lambda_0 \\mathbb{E}[f_{12}]^2 \\| {\\boldsymbol x}^2 \\|^4$. To prove the BBP transition, we can proceed as in the current paper with several changes. The most significant change is that the noise matrix $V$ in (14) is now given by $$V_{ij} = \\frac{f_{ij}}{\\sqrt N} + \\sqrt{\\lambda_0} N^{1/4} f_{ij}' x_i x_j + \\sqrt{\\lambda_0} N (f_{ij}''- \\mathbb{E}[f_{ij}'']) x_i^2 x_j^2.$$ Due to the change of the noise matrix $V$, we need to adjust several parts of the current paper. The most notable ones are as follows:\n\n(1) The (row) sum of the variances, $\\sum_j \\mathbb{E}[(V_{ij})^2] = 1+ \\Theta(N^{-1/2})$ (which was $1+ \\Theta(N^{-1})$). This means that all eigenvalues are rescaled by a (deterministic) factor $1+ \\Theta(N^{-1/2})$, or equivalently, all eigenvalues are shifted by $\\Theta(N^{-1/2})$. Since the shift is not negligible when compared to the typical size of the fluctuations ($\\Theta(N^{-1/2})$ or $\\Theta(N^{-2/3})$), our main results should be changed accordingly. \n\n(2) The proof of the local law for $V(t)$, Lemma B.3, and other proofs based on the local law and the resolvent identity, will become more involved. We need to use the bootstrap argument more times, where we first prove a (weak) estimate first and improve it by using the weak estimate itself. \n\nIn the revision, we will incorporate the discussion above into the manuscript and add detailed proofs to the appendix.", "claims_and_evidence": "The claims made in the paper are well-supported by rigorous mathematical proofs. I have checked --not in detail-- the theoretical derivations and found them correct.", "methods_and_evaluation_criteria": "Since this is a theoretical work, there are no experimental design concerns. The supplementary material, which contains additional proofs, was reviewed.", "theoretical_claims": "The methodology follows standard theoretical techniques in random matrix theory, and the supplementary material provides complete proofs of the results.", "experimental_designs_or_analyses": "N/A", "relation_to_broader_scientific_literature": "The paper builds on the well-established literature on spiked random matrix models and eigenvalue phase transitions, particularly the Baik–Ben Arous–Péché (BBP) transition. It extends results from classical spiked Wigner matrix theory by considering entrywise transformations and their impact on spectral properties, a direction that has been explored in works such as Lesieur et al. (2015), Perry et al. (2018).\n\nThe main novelty here is the rigorous characterization of eigenvalue fluctuations under such transformations, demonstrating that the largest eigenvalue follows a Gaussian or Tracy–Widom GOE distribution depending on the effective signal strength. This aligns with recent advances in random matrix theory that analyze the stability and universality of spectral methods under transformations, as seen in both theoretical works and applications in statistical inference and machine learning.\n\nWhile the paper is mathematically strong, its direct connection to broader ML applications is somewhat limited. Similar ideas have been used in theoretical analyses of feature learning in neural networks and optimization landscapes, suggesting a potential impact beyond pure RMT. The author discusses some of these aides, but  making this connection more explicit could strengthen its relevance to the ICML audience?", "essential_references_not_discussed": "The literature review in this paper is well done, and I do not have much to add. The only minor but possible suggestion is related to a question bellow is related to  arXiv:2403.04234, \"Fundamental Limits of Non-Linear Low-Rank Matrix Estimation.\"", "other_strengths_and_weaknesses": "The paper is solid and presents a rigorous mathematical analysis of eigenvalue fluctuations in transformed spiked Wigner matrices. However, it leans heavily towards mathematical random matrix theory (RMT) rather than machine learning, which raises the question of its relevance to the ICML audience. While the results are interesting and technically sound, their direct impact on practical ML applications remains somewhat unclear. The theoretical insights are valuable for understanding spectral methods, but their significance for ICML could be better motivated.", "other_comments_or_suggestions": "See bellow", "questions_for_authors": "My main question concerns a possible generalization. In works such as Guionnet et al. (2023) and Feldman (2023), a slightly different scaling is considered, which is further explored in detail in arXiv:2403.04234.  In this generalization, the transformation starts with  a signal-to-noise ratio scaling as N^{1-1/k_f) , but still ends up with a BBP-type transition for the trasnformed spectral method.  I am curious whether the authors' approach could also yield Tracy–Widom-type fluctuations in this setting. If so, this would significantly enhance the impact and applicability of the results.", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {"ethical_review_concerns": "N/A"}}, {"review_id": "4tQNq4QiXn", "reviewer": "Reviewer_kzF3", "summary": "The authors study the largest eigenvalue of a spiked Wigner matrix model under elementwise function transformation. They show that the largest eigenvalue of the transformed matrix undergoes a phase transition when an effective SNR variable is tuned. At high SNR, the find that the largest eigenvalue is distributed as a gaussian around a value computable from the effective SNR. At low SNR, it is a Tracy-Widom distribution around 2. The authors establish this via proof and support their claims via numerics.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for the constructive review and helpful feedback with which we can greatly improve our paper. Below, we address the concerns and questions, and also outline some important revisions we will make.\n\n**Experimental design**: Thank you for your comment. We will include the graphs of the density $p$ and the transformation $f$ in the revision. (The precise formulas for $p$ and $f$ can be found in Appendix A.1.) See also the following links for the graphs of the [density](https://www.wolframalpha.com/input?i=Draw+%282.5%2F%5Csqrt%7B2%5Cpi%7D%29+*%5Be%5E%7B-%285%2F2%29%28x-2%2Fsqrt%7B5%7D%29%5E2%7D+%2B+e%5E%7B-%285%2F2%29%28x%2B2%2Fsqrt%7B5%7D%29%5E2%7D%5D) and the [transformation](https://www.wolframalpha.com/input?i=Draw+2.62503*%5B%28x-2%2F%5Csqrt%7B5%7D%29+e%5E%7B-%285%2F2%29%28x-2%2Fsqrt%7B5%7D%29%5E2%7D+%2B+%28x%2B2%2F%5Csqrt%7B5%7D%29+e%5E%7B-%285%2F2%29%28x%2B2%2Fsqrt%7B5%7D%29%5E2%7D%5D%2F%5B+e%5E%7B-%285%2F2%29%28x-2%2Fsqrt%7B5%7D%29%5E2%7D+%2B++e%5E%7B-%285%2F2%29%28x%2B2%2Fsqrt%7B5%7D%29%5E2%7D%5D).\n\n**Other comments and questions**: \n\n(1)-(3) Thank you for pointing out the typos. We will correct them and other typos in the revision. Also, $f$ in the definition in $\\lambda_e$ is a typo, which should be $f'$.\n\n(4) If $\\lambda_e = 1$, then the largest eigenvalue $\\mu_1 (\\widetilde{M})$ converges to $2$, but the fluctuation $N^{2/3}( \\mu_1 (\\widetilde{M}) - 2)$ converges to a distribution different from the GOE Tracy-Widom distribution. To our best knowledge, the limiting distribution involves the stochastic Airy operator and does not have simple formula. We refer to Theorems 1.5 and 1.7 in (Bloemendal and Virag, 2013) for more detail.\n\n(5) If we let $y_i$ be i.i.d. with an $N$-independent distribution whose mean is $0$, variance is $1$, and all moments are finite, and set $x_i = y_i/\\sqrt{N}$, then Assumption 2.3 is satisfied with overwhelming probability. (This can be proved by concentration inequalities such as a high-order Markov inequality.) Our assumption on the spike is not much different from the assumptions in other related papers, e.g., (Guionnet et al., 2023; Feldman, 2023). Further, the prior distribution may have an atom at $0$, so the spike does not need to be 'dense' or 'flat'. If we consider the community detection case, Assumption 2.3 is satisfied if $|\\{ x_i = 1/\\sqrt{N} \\}| - |\\{ x_i = -1/\\sqrt{N} \\}| = O(N^{1/2 + \\epsilon})$, which implies that the model is (almost) balanced.\n\n(6) If we let $\\boldsymbol v$ be the eigenvector corresponding to the largest eigenvalue, it is known that $\\langle \\boldsymbol v,  \\boldsymbol x^2 \\rangle \\to (1 -1/\\lambda_e)$ if $\\lambda_e > 1$, and $\\langle \\boldsymbol v, \\boldsymbol x^2 \\rangle \\to 0$ otherwise. (See Corollary 2.2 in (Guionnet et al., 2023).)", "claims_and_evidence": "Yes, the claims are well supported", "methods_and_evaluation_criteria": "Yes, the proposed methods make sense.", "theoretical_claims": "I checked the proof of the main theorems. Propositions were not checked.", "experimental_designs_or_analyses": "Yes, I checked the validity of both the numerical experiments.\nThe first numerical experiment is not very clear to me. The f used here is defined in terms of the p and it's derivative. The authors should be more concrete about what this function explicitly looks like.", "relation_to_broader_scientific_literature": "The paper basically shows what the statistical fluctuations of the largest eigenvalue looks like in the spiked Wigner model under element wise transformations. In the un-transformed case,  similar results are known and have applications in community detection. This is a useful generalization of this as it allows us to rigorously boost the SNR in these problems.", "essential_references_not_discussed": "N/A", "other_strengths_and_weaknesses": "See questions", "other_comments_or_suggestions": "1.) After eq(1) there is a typo. H should be W \n\n2.) In Eq(4) the bracket in f' is not closed\n\n3.) In several points in the paper \\lambda_e is defined with f instead of f'. This looks like a typo. If defined with f, the effective SNR will vanish by assumption.\n\n4.) For the sake of completeness, could the authors briefly discuss what happens when \\lambda_e = 1\n\n 5.) How are the assumptions 2.3 on x justified? It seems to say that the x vector must be fairly flat. For the community detection case where x_i are proportional to {-1,1} what do these assumptions mean? Does it translate to a statement about the size of the communities?\n\n6.) Do the techniques used here tell us something about the eigenvector corresponding to the largest eigenvalue? For PCA type of applications this is important to know. Will this eigenvector have a large overlap with the spike?", "questions_for_authors": "N/A (See previous)", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "rbriODieUU", "reviewer": "Reviewer_RXhe", "summary": "This work investigates the asymptotic properties of the largest eigenvalue of entry-wise, non-linear transformations of spiked Wigner matrices. Its main result is to prove a BBP-like result: below a certain critical SNR (which is explicitly given in terms of the original SNR), the top eigenvalue has asymptotic Gaussian fluctuations, while below it it follows a Tracy-Widom law. The result is proven using an interpolation + resolvent comparison argument that requires proving a local law for this class of matrices, which is of independent interest. Numerical simulations are also given to illustrate the result.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for the constructive review and helpful feedback with which we can greatly improve our paper. Below, we address the concerns and questions, and also outline some important revisions we will make.\n\n**References**: Thank you for the references that we missed. We will include them in the revision.\n\n**Relevance to the ICML audience**: Thank you for the comment. In the revision, we will add and change a few paragraphs in the introduction of the current manuscript to demonstrate the relevance of our model to the ICML audience. The details can be found in our answer to Reviewer 1 (dwXH); I am sorry that we cannot write all the details in this box due to the length limit,  \n\n**Other comments and questions**: \n\n(1) Thank you for pointing out the typos. We will correct them and other typos in the revision. Also, $f$ in the definition in $\\lambda_e$ is a typo, which should be $f'$.\n\n(2) Thank you for your comment on the large SNR regime. If $\\lambda_e = \\omega(1)$, we believe that our result on the supercritical case still holds, i.e., $N^{1/2}(\\mu_1 - \\sqrt{\\lambda_e} - 1/\\sqrt{\\lambda_e})$ converges to a centered Gaussian distribution with variance $2$. However, to prove it rigorously by adapting the strategy in the current manuscript, we need to find a proper target matrix for which the limiting distribution of the largest eigenvalue is proven, but unfortunately, to our best knowledge, it is not proven. Nevertheless, we would remark that the large ($\\omega(1)$) SNR case should be relatively easier to analyze than the $O(1)$ SNR case and the argument in the current manuscript should be applicable to this case without many changes.\n\nWe also would remark that one can consider the case where the SNR is large but $\\lambda_e = 0$ since $\\mathbb{E}[f_{12}'] = 0$. See our answer to Reviewer 1 (dwXH), item **Question on generalization**, for more detailed discussion on this case.\n\n(3) The main technical challenge for the non-symmetric or rectangular case is that the local law is much harder to prove. More specifically, we need to use a self-consistent equation for the resolvent entries analogous to the quadratic vector equation (21), but its analysis is significantly more complicated due to some technical issues. Furthermore, if we consider a multiplicative model considered in (Perry et al., 2018; Jung et al., 2021), then the transformed matrix cannot be well approximated by a spiked random matrix. This means that the strategy in the current manuscript should be changed accordingly.\n\n(4) If we let $y_i$ be i.i.d. with an $N$-independent distribution whose mean is $0$, variance is $1$, and all moments are finite, and set $x_i = y_i/\\sqrt{N}$, then Assumption 2.3 is satisfied with overwhelming probability. The prior distribution (of $y_i$) does not need to be compactly supported, so our assumption on the spike is weaker than that in (Mergny et al. 2024) in this sense. Further, the prior distribution may have an atom at $0$, so the spike does not need to be 'dense' or 'flat'. \n\nOne possible generalization of our result is to allow a `sparse' spike in the sense that $\\max_i |x_i| > N^{-1/2 +\\delta}$. With such a sparse spike, the fluctuation of the largest eigenvalue might exhibit another phase transition depending on $\\delta > 0$. This is an interesting topic and we would like to consider it in the future.", "claims_and_evidence": "Yes, all claims in this work are rigorous mathematical statements, and the proofs are provided.", "methods_and_evaluation_criteria": "N/A.", "theoretical_claims": "I skimmed through the proof, and did not spot any issue. The arguments are standard, but require controlling a rank-2 spiked Wigner matrix.", "experimental_designs_or_analyses": "N/A.", "relation_to_broader_scientific_literature": "The universality of entry-wise non-linear functions of spiked Wigner matrices is the topic of several previous works, since it has important consequences for statistical inference. Different flavours of this result exist in the literature, the one most relevant to this work being Perry et al., (2018), which this work build upon and is thoroughly acknowledged.\n\nHowever, some relevant important literature is missing - see below.", "essential_references_not_discussed": "To my best knowledge, the first rigorous universality results in this direction appeared in (Krzakala et al., 2016; Dia et al., 2016; Lelarge & Miolane 2017), who proved the universality of the asymptotic mutual information with respect to the noise channel conjectured by Lesieur et al., (2015). These are key related reference from which (Perry et al., 2018) build upon.\n\nMore recently, (Mergny et al. 2024) extended these universality results to cases in which the Fisher information is zero, requiring an analysis of different SNR regimes.\n\nOn the related literature side, a recent result by Dandi et al., (2024) proved rigorous deterministic equivalents for non-linear spiked models coming from single gradient steps.\n\n- [Krzakala et al., 2016] Krzakala, Florent, Jiaming Xu, and Lenka Zdeborová. \"Mutual information in rank-one matrix estimation.\" In 2016 IEEE Information Theory Workshop (ITW), pp. 71-75. IEEE, 2016.\n\n- [Dia et al., 2016] Dia, Mohamad, Nicolas Macris, Florent Krzakala, Thibault Lesieur, and Lenka Zdeborová. \"Mutual information for symmetric rank-one matrix estimation: A proof of the replica formula.\" Advances in Neural Information Processing Systems 29 (2016).\n\n- [Lelarge & Miolane 2017] Lelarge, Marc, and Léo Miolane. \"Fundamental limits of symmetric low-rank matrix estimation.\" In Conference on Learning Theory, pp. 1297-1301. PMLR, 2017.\n\n- [Mergny et al. 2024] Mergny, Pierre, Justin Ko, Florent Krzakala, and Lenka Zdeborová. \"Fundamental limits of non-linear low-rank matrix estimation.\" Proceedings of Thirty Seventh Conference on Learning Theory, PMLR 247:3873-3873, 2024.\n\n- [Dandi et al., 2024] Dandi, Yatin, Luca Pesce, Hugo Cui, Florent Krzakala, Yue M. Lu, and Bruno Loureiro. \"A random matrix theory perspective on the spectrum of learned features and asymptotic generalization capabilities.\" arXiv preprint arXiv:2410.18938 (2024).", "other_strengths_and_weaknesses": "Overall, the paper is well-written and easy to read, and the result is solid. However, I wonder about the choice of venue, since as a \"pure\" RMT paper it will only interest a small part of the ICML community. Maybe the authors could have put more effort into motivating how the spiked estimation problem pops up in ML, giving some more concrete examples and context. Otherwise, I think it is a good RMT paper.", "other_comments_or_suggestions": "- Page 1, below eq. (1), $H$ should read $W$.", "questions_for_authors": "- In the first paragraph of \"1.1. Contributions\" and in eq. (5), should the effective SNR be $f'$ instead? Maybe I am missing something, but if this is correct, how do you reconcile with the result of (Lesieur et al., 2015) discussed in eq. (3)?\n\n- On the ML side, there has been a wave of recent interest in non-linear transformations of spiked Wigner/Wishart matrices, mostly motivated by the fact that, asymptotically, a single step of GD for a two-layer neural network trained with the square loss looks like a matrix in this ensemble. This characterization was used to prove the benefit of feature learning in several works (Ba et al. 2022; Moniri et al., 2023; Cui et al., 2024; Dandi et al., 2024).\n\n  However, an important point in these works is that feature learning is only beneficial when the scaling of the spike (SNR here) is very large - i.e. scales with $N$. This regime is way above the BBP transition. Would the theorem here apply? If not, could the argument be generalized to cover this case?\n\n- What are the main technical challenges to extend the proof to the non-symmetric, spiked Wishart case?\n\n- How important is the assumption that the spike is dense $||x||_{2}^{2}=1$? The universality proofs for the mutual information of [Krzakala et al. 2016; Mergny et al. 2024] only require the support of the spike distribution to be finite.", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Meta-review: This paper provides a detailed fluctuation analysis of the largest eigenvalues in spiked Wigner matrices under entrywise nonlinear transformations. The main technical contribution lies in identifying the non-universality of the leading eigenvalue fluctuations in this setting—showing that unlike classical spiked models, the fluctuations depend on the specific form of the transformation function. The reviewers are in broad agreement that this is a valuable contribution to the theory of random matrices with relevance for understanding the behavior of spectral methods in high-dimensional settings. The paper is technically strong and carefully written.

At the same time, reviewers noted that the presentation is somewhat dense and could be improved to make the significance of the results more accessible. In particular, it would help to provide stronger motivation from machine learning contexts where such transformations naturally arise—e.g., in kernel PCA, graph embeddings, or score-based generative models. Similarly, adding a clearer explanation of how the fluctuation results relate to algorithmic questions (e.g., detectability thresholds, inference error) would broaden the paper’s impact. While the results are mathematical, the framing could better bridge the gap to applications that ICML readers care about. The authors are encouraged to clarify these points in the final version. Given the novelty and technical depth, the paper deserves to be accepted.
Decision: Accept (poster)