Freeness over the diagonal and outliers detection in deformed random matrices with a variance profile
J\'er\'emie Bigot, Camille Male
TL;DR
This paper analyzes the eigenvalue and singular value distributions of deformed Gaussian matrices with variance profiles, using free probability and deterministic equivalents, to detect outliers and improve low-rank denoising in heteroscedastic noise.
Contribution
It introduces a fixed point equation approach based on freeness with amalgamation to approximate spectral distributions and identify outliers in deformed random matrices with variance profiles.
Findings
Derived a non-asymptotic fixed point equation for spectral distribution approximation.
Characterized outlier locations in deformed GUE matrices with variance profiles.
Demonstrated application to low-rank matrix denoising under heteroscedastic noise.
Abstract
We study the eigenvalue distribution of a GUE matrix with a variance profile that is perturbed by an additive random matrix that may possess spikes. Our approach is guided by Voiculescu's notion of freeness with amalgamation over the diagonal and by the notion of deterministic equivalent. This allows to derive a fixed point equation to approximate the spectral distribution of certain deformed GUE matrices with a variance profile and to characterize the location of potential outliers in such models in a non-asymptotic setting. We also consider the singular values distribution of a rectangular Gaussian random matrix with a variance profile in a similar setting of additive perturbation. We discuss the application of this approach to the study of low-rank matrix denoising models in the presence of heteroscedastic noise, that is when the amount of variance in the observed data matrix may…
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
\WarningFilter
latexMarginpar on page
Freeness over the diagonal and outliers detection in deformed random matrices with a variance profile
Jérémie Bigot & Camille Male
Institut de Mathématiques de Bordeaux et CNRS (UMR 5251)
Université de Bordeaux
Abstract
We study the eigenvalue distribution of a GUE matrix with a variance profile that is perturbed by an additive random matrix that may possess spikes. Our approach is guided by Voiculescu’s notion of freeness with amalgamation over the diagonal and by the notion of deterministic equivalent. This allows to derive a fixed point equation to approximate the spectral distribution of certain deformed GUE matrices with a variance profile and to characterize the location of potential outliers in such models in a non-asymptotic setting. We also consider the singular values distribution of a rectangular Gaussian random matrix with a variance profile in a similar setting of additive perturbation. We discuss the application of this approach to the study of low-rank matrix denoising models in the presence of heteroscedastic noise, that is when the amount of variance in the observed data matrix may change from entry to entry. Numerical experiments are used to illustrate our results.
Keywords: Deformed random matrix, Variance profile, Outlier detection, Free probability, Freeness with amalgamation, Operator-valued Stieltjes transform, Gaussian spiked model, Low-rank model.
Contents
-
1.1.3 Global behavior of the eigenvalues of an additive perturbation
-
2.1 Additive perturbations and the standard spiked population model
-
2.3 Random matrices with a variance profile in free probability
-
6.3 Stability of the fixed point equation and proof of Lemma 4.1
1 Introduction
We first introduce some questions related to the estimation of the eigenvalue distribution of the sum of a GUE matrix with a variance profile and a deterministic matrix that may possess spikes. Then, we present the main theoretical contributions of the paper and its organization.
1.1 Aim of the article
1.1.1 Deformed Hermitian random matrices
We recall that for any integer , a GUE matrix is a Hermitian random matrix \big{(}\frac{x_{i,j}}{\sqrt{N}}\big{)}_{i,j=1,\ldots,N}, such that the sub-diagonal entries are independent, the variables , , are centered real Gaussian variables with variance one, and the non-diagonal entries (with ) are centered complex Gaussian variables with variance one. Let \Gamma_{N}=\big{(}\gamma_{N}^{2}(i,j)\big{)}_{i,j=1,\ldots,N} be a symmetric matrix with non negative entries. Then, the random matrix
[TABLE]
is called a GUE matrix of size with variance profile . Furthermore, we consider a deterministic Hermitian matrix . The random matrix
[TABLE]
is then an (additively) deformed random matrix. We shall also study the situation where has spikes that may generate a finite number of eigenvalues (called outliers) that detach from the rest of the spectrum.
There are two dual ways to interpret model (1.2) in RMT. From the point of view of mathematical physics, the random matrix models the Hamiltonian of a system, and the deformation is a perturbation coming from an external source. A variance profile for can model impurities of the system. From the signal processing point of view, models a signal and an additive noise which deforms the data.
In this paper, we present results and heuristics to approximate the eigenvalue distribution of a deformed random matrix by a deterministic function, called classically a deterministic equivalent. Our approach involves tools in RMT and free probability, some of them having been created quite recently. They may have applications for the numerical analysis of the spectral distribution of data organized in a matrix form such as noisy images.
The substance of our mathematical arguments is a legacy of the investigations of Haagerup and Thorbjørnsen who developed a strategy in [37] for the study of block GUE matrices, involving classical techniques of Gaussian calculus and complex analysis. In their seminal paper, such a strategy served as an ingredient to reinforce the link between RMT and the abstract theory of -algebras of free groups. The analysis in [37] and in our work are both based on Voiculescu’s equations of subordination with amalgamation. Adapting Haagerup and Thorbjørnsen’s technique, we want to spread their method to more applied contexts that we believe to be of interest in high-dimensional statistics and signal processing. We notify the reader that the aim of this work is not to obtain sharp estimations (that is with optimal rates of convergence) of the spectral distribution of deformed random matrix with a variance profile, and we refer to [30] for recent results on this topic.
1.1.2 Information plus noise model
If model (1.2) is quite natural from the mathematical point of view, it is more interesting for application in statistics to study rectangular matrices with no symmetry. We define a standard Gaussian matrix as a rectangular random matrix \big{(}\frac{x_{i,j}}{\sqrt{M}}\big{)}_{i,j} with independent centered complex Gaussian entries with variance one for all . Let \Gamma_{N,M}=\big{(}\gamma_{N,M}^{2}(i,j)\big{)}_{i,j} be a matrix with non negative entries. Then, the random matrix
[TABLE]
is called a Gaussian matrix with variance profile . Let be a deterministic matrix of size . The random matrix
[TABLE]
is called an information plus noise model with a variance profil.
When is a finite rank matrix, then the rectangular information plus noise model (1.3) corresponds to the low-rank matrix denoising problem which arises in various applications, where it is of interest to estimate a by signal matrix from noisy data. When the variance profile is a matrix with equal entries, the problem of estimating the low-rank matrix has been extensively studied in statistics and machine learning [55, 19, 26, 47] using spectral estimators constructed from the singular value decomposition of . These works build upon well understood results of the asymptotic behavior (as ) of the singular values of in the Gaussian spiked population model [14, 27].
Hence, low-rank matrix estimation is well understood when the additive noise is Gaussian with homoscedastic variance (that is a constant variance profile). However, in many applications the noise can be highly heteroscedastic meaning that the amount of variance in the observed data matrix may significantly change from entry to entry. Examples can be found in photon imaging [53], network traffic analysis [11] or genomics for microbiome studies [20]. In such applications, the observations are count data that are modeled by Poisson or multinomial distributions which leads to heteroscedasticity. The literature on statistical inference from high-dimension matrices with heteroscedastic noise has thus recently been growing [16, 43, 58, 62, 32].
As discussed in [62], the Gaussian model (1.3) with a variance profile can serve as a prototype for various applications involving low-rank matrix denoising in the presence of heteroscedasticity. The analysis of the outliers in model (1.3) with a non-constant variance profile is also of interest in neural networks to model synaptic matrix [52] and in signal processing for radio communications [24]. Nevertheless, to the best of our knowledge, the characterization of potential outliers in model (1.3) when the noise matrix has a variance profile that is not constant has not received very much attention so far.
We now address the two questions on deformed random matrices we investigate.
1.1.3 Global behavior of the eigenvalues of an additive perturbation
The first issue is the global behavior of the eigenvalues of . In particular, under general assumptions on and , we shall consider the question of how approximating the empirical spectral distribution (e.s.d.) of :
[TABLE]
where is the Dirac mass and the -th eigenvalue of . As commonly done in random matrix theory (RMT), we use the Stieltjes transform of the e.s.d. of , namely the map defined by the trace of the resolvent of that is
[TABLE]
In above formula, belongs to the set of complex number with positive imaginary part. When the context is clear, we write for the matrix . This transform has numerous properties and applications, we refer to Benaych-Georges and Knowles lecture notes [13, Section 2.2] for an introduction. Note that for (with and ) then where denotes the convolution product and is an approximate delta function (Cauchy kernel) as . Hence for chosen sufficiently small, the function
[TABLE]
is a good approximation of over by a random analytic function. In what follows, the smooth density defined by (1.5) will be referred to as the inverse Stieltjes transform of .
Then, we first address the question of the construction of a deterministic analytic function , which only depends on , on the variance profile and on , and that approximates with high probability.
1.1.4 Outliers location in finite rank perturbation models
The second issue is the effect of a low rank additive perturbation. For notational convenience, we introduce another deterministic Hermitian matrix of finite rank (not depending on ). We can thus write , where is a diagonal matrix with nonzero diagonal entries and is a matrice whose columns are orthonormal vectors. The eigenvalues of are commonly called the spikes. The matrix
[TABLE]
is a finite rank deformation of . For such models, we consider the problem of how locating in the spectrum of the eigenvalues coming from that detach from the spectrum of .
To formulate more precisely this problem, we say that, for a given level of precision , an outlier is a real eigenvalue of which is not in an -neighborhood of the spectrum of . Following [14], let us first factorize the spectrum of in , in the following sense. For any complex which is not in the spectrum of , we consider the decomposition
[TABLE]
where
[TABLE]
An outlier is then a real number away from the spectrum of such that \det\big{(}\alpha_{N}(t)\big{)}=0. Therefore, noticing that the determinant of is equal to the determinant of the matrix
[TABLE]
it follows that, to localize the outliers of , it is sufficient to compute the real numbers such that
[TABLE]
Then, the second question that we address in this paper is the construction of a deterministic matrix-valued function , which depends on , on the variance profile , and on the matrices , and that approximates . In this manner, the zeros of that are away from the spectrum of shall indeed be close to the outliers of .
1.2 Main statements
We now provide the formal statements of our main results and the method to answer the two questions raised above. We let (resp. ) denote the set of diagonal matrices \Lambda=\big{(}\Lambda(i,j)\big{)}_{i,j} of size with diagonal entries having positive imaginary (resp. negative) parts. For any matrix , we denote by the diagonal matrix whose diagonal entries are those of .
The operator-valued Stieltjes transform of a Hermitian matrix is the map
[TABLE]
For any , we also introduce the mapping
[TABLE]
that is a key tool in our analysis. The map may also be written as
[TABLE]
where for a matrix is the diagonal matrix whose -diagonal element is the sum of the entries of the -row of . We now state our main result on the construction of a deterministic equivalent of the operator-valued Stieltjes transform of deformed random matrices and its finite sample properties of approximation.
Theorem 1.1**.**
There exists a unique function , analytic in each variable, that solves of the following fixed point equation
[TABLE]
for any . Let , let , and consider satisfying
[TABLE]
Then, for any , setting
[TABLE]
we have, for ,
[TABLE]
where denotes the operator norm of a matrix.
If moreover is diagonal, for any satisfying
[TABLE]
then (1.14) holds with
[TABLE]
The solution of the fixed point equation is referred to as the deterministic equivalent of the operator-valued Stieltjes transform of . It has another description given in Section 6.2.2, as the limit of a functional on large random matrices. One may remark that the results of Theorem 1.1 hold without any assumption on the Hermitian matrix and the variance profile . In particular no bound from below for the entries is involved to derive the concentration inequality (1.14). If we are given sequences of matrices with growing dimension , then this estimate is also meaningful when grows slowly with .
The proof of Theorem 1.1 is divided into three steps that are detailed in Section 5, Section 6 and Section 7. We then deduce the following methods to answer the two issues raised in Section 1.1.3 and Section 1.1.4.
Approximation of the global behavior of the e.s.d. of by a smooth density.
The proof of Theorem 1.1. with a slightly different argument of concentration implies the following approximation for the Stieltjes transform of (recall that ).
Corollary 1.2**.**
For any such that
[TABLE]
denoting for
[TABLE]
and , it follows that
[TABLE]
where is the unique solution of (1.12) for the scalar matrix . If moreover is diagonal and , then (1.17) holds with
[TABLE]
The concentration inequality (1.17) means that the deterministic equivalent is a good approximation of when the imaginary part of is allowed to decay to zero as the dimension grows at a rate given by the right hand side of Inequality (1.16). Hence, by the inverse Stieltjes transform formula (1.5), the map is a good approximation for the e.s.d. of when is small. However, compared to existing results in the RMT literature, the lower bound (1.16) on is not optimal, and thus Corollary 1.2 can only be interpreted as a weak local law. More generally, the lower bound condition for in Theorem 1.1 means that all diagonal entries of must have an imaginary part that is sufficiently large as shown by the Conditions (1.13) and (1.15). Therefore, Theorem 1.1 may be interpreted as an operator-valued weak local law for the operator-valued Stieltjes transform of . In Section 2.4, we discuss more precisely the connections between our work and existing results on local laws for random matrices with a variance profile.
Outliers localization in the case where is diagonal.
From its definition, the natural way to construct a deterministic equivalent for is to replace in expression (1.7) by an appropriate deterministic estimate. When is diagonal, then for any , the matrix \mathbb{E}\big{[}(\Lambda-X_{N}-Y_{N})^{-1}\big{]} is also diagonal (thanks to Corollary 7.2). Using a concentration inequality, one has that is thus close to a diagonal matrix, and so we can replace this generalized resolvent with the deterministic equivalent of the operator-valued Stieltjes transform.
Corollary 1.3**.**
We assume that is diagonal and define the deterministic matrix valued-function function
[TABLE]
Then, for any such that , then is a deterministic equivalent of , in the sense that
[TABLE]
where, for any ,
[TABLE]
The proof of Inequality (1.19) can be found in Section 7.
Outliers localization, general case.
In general, if is not diagonal then the expectation of the generalized resolvent, that is \mathbb{E}\big{[}(\Lambda-X_{N}-Y_{N})^{-1}\big{]}, is no longer a diagonal matrix. Hence it is no longer correct to approximate by replacing with in Equation (1.7). Yet, it is proved in Section 5 that the generalized resolvent can be approximated by the following deterministic matrix
[TABLE]
with \Omega_{X_{N},Y_{N}}(\Lambda):=\Lambda-\mathcal{R}_{N}\big{(}\mathbb{E}\big{[}G_{X_{N}+Y_{N}}(\Lambda)\big{]}\big{)}. It appears that the matrix can be interpreted as an approximate operator-valued subordination function, and we refer to Sections 2.2 and 2.3 for further details and discussion on this heuristic.
Corollary 1.4**.**
We define the deterministic matrix valued-function function
[TABLE]
where
[TABLE]
with solution of the fixed point Equation (1.12). Then, for any such that , then is a deterministic equivalent of , in the sense that
[TABLE]
where, for any ,
[TABLE]
Remark 1.5*.*
A key argument developed to obtain the proof of Inequality (1.22) is the derivation of an upper bound on the difference in operator norm between \mathbb{E}\big{[}(\lambda\mathbb{I}_{N}-X_{N}-Y_{N})^{-1}\big{]} and its approximation by \big{(}\Omega_{H_{N}}^{\square}(\lambda\mathbb{I}_{N})-Y_{N}\big{)}^{-1}. In the general case, when is held fixed, this error term decays at the rate whereas, in the case where is diagonal, the operator norm of the difference between \mathbb{E}\big{[}(\lambda\mathbb{I}_{N}-X_{N}-Y_{N})^{-1}\big{]} and decays at the rate . Note that an integrable decay such as , , implies the convergence of the spectrum also called strong convergence [46]: almost surely for large enough the spectrum of belongs to a small neighborhood of the support of the measure whose Stieltjes transform is . Under mild assumptions, the strong convergence of is proved in [30, Corollary 2.3] with possibly non diagonal.
Remark 1.6*.*
In general, even when the locations of outliers possibly depend on the eigenvectors of , and there is no longer a canonical way to associate an outlier to a specific spike as it is the case when the variance profile has constant entries. We shall discuss this specific case in the literature review proposed in Section 2, and this property will be illustrated by numerical experiments in Section 3.
1.3 Organization of the paper
In Section 2 we relate the approach followed in this paper to the existing literature in RMT and free probability on deformed models and outlier detection. Then, we report the results of numerical experiments in Section 3 to shed some light on the benefits of our approach to localize potential outliers in information plus noise models with a variance profile. We present the strategy of the proof of Theorem 1.1 and its organization in Section 4. The details of the main steps of the proof are then gathered in Section 5, Section 6 and Section 7.
Acknowledgements.
Jérémie Bigot is a member of Institut Universitaire de France (IUF), and this work has been carried out with financial support from the IUF. Camille Male received the support of the Simons CRM Scholar-in-Residence Program. We would like to also thank László Erdös for his remarks on a previous version of this paper and for pointing out recent references.
1.4 Publicly available source code
For the sake of reproducible research, Python scripts available at the following address https://www.math.u-bordeaux.fr/~jbigot/Site/Publications_files/ScriptsSpikesVarianceProfile.zip allow to reproduce the numerical experiments carried out in this paper.
2 Related literature and methods
2.1 Additive perturbations and the standard spiked population model
We first discuss the case when all entries of the variance profile equal one, that is is a standard GUE matrix. The celebrated Wigner’s Theorem [61] states that the empirical spectral distribution of converges to the semicircular distribution . Pastur studies in [49] the global behavior of a GUE matrix with additive perturbation. Assuming that the e.s.d. of converges to a measure , then almost surely the e.s.d. of converges to a measure denoted . In general, there is no explicit description of , but the limiting Stieltjes transform of satisfies the so-called Pastur’s equation, which is expressed in terms of the Stieltjes transform of as follows: for all in , we have
[TABLE]
Note that implies and the above equation reads g_{x}(\lambda)=\big{(}\lambda-g_{x}(\lambda)\big{)}^{-1}.
In practical applications, we are given a matrix of fixed size and not a sequence whose e.s.d. converges to a certain distribution . The deterministic equivalent method, for Pastur’s equation (2.1), consists in replacing by the true e.s.d. . In this setting, there is a unique analytic map , solution of the fixed-point equation
[TABLE]
for a map . Note that there is a convenient abuse of notation in the sense that depends only on . We say that is a deterministic equivalent of . The interest is that is a good approximation of of the Stieltjes transform that we can approximate numerically thanks to the fixed-point equation (2.2).
A fundamental result of phase transition for finite rank deformed random matrices was discovered by Ben Arous, Baik and Péché [10] in the slightly different model of Gaussian matrices without symmetry. Called in short BBP-transition, the analogue result for GUE matrices [51] states that, in the large limit, a spike of will create an outlier in only if in which case . For related results, we also refer to [34, 33, 22, 23].
Remark 2.1*.*
When all entries of the variance profile equal one, we emphasis that the existence and the position of an outlier is independent of the eigenvector of associated to the spike .
2.2 The free probability approach
More generally, the issues discussed above have also been considered when is a unitary invariant random matrix. In this context, Voiculescu’s notion of asymptotic freeness [60] implies a generalization of Pastur’s equation. Assume that and have limiting e.s.d. and respectively. Recall that, denoting the Stieltjes transform of , the -transform of is the analytic map satisfying
[TABLE]
Then converges to a measure , called the free convolution of and . This limit is characterized by the so-called subordination property: ,
[TABLE]
The map is linear if and only if is a centered semicircular distribution. The method of deterministic equivalent can be extended to this case using Voiculescu’s equation (2.4) instead of Pastur’s one. The difficulty in general is to compute the -transform , or to replace it with a good approximation.
For finite rank deformation, an important discovery was made in the early decade by Capitaine [21]. We recall the heuristic presented in [12] and refer to this paper for the mathematical arguments, without defining the notions of free probability. In the context of Voiculescu’s problem, the limit of and of are modeled in the free von Neumann algebra generated by two self-adjoint variables and with distribution and respectively. Let be the projection on the von Neumann algebra generated by . Then Biane proved [15] that there exists an analytic map defined outside the spectrum of , called the subordination function, such that
[TABLE]
The above equality means that the projection of the resolvent of equals the resolvent of evaluated at the subordination function. Taking the trace in the identity yields the relation for Stieltjes transforms
[TABLE]
The reader not familiar with free probability language can still translate this result into a heuristic for unitary invariant matrices by replacing the condition expectation by the classical expectation and the variables by the matrices. One method to introduce an approximate subordination function consists in setting
[TABLE]
so that we have \mathbb{E}\big{[}(\lambda-X_{N}-Y_{N})^{-1}\big{]}=\big{(}\omega_{X_{N},Y_{N}}(\lambda)-Y_{N}\big{)}^{-1}, similar to (2.5).
The matrix is not a scalar, but this property is true for the compression involved in the outlier detection problem, in the following sense. If is unitarily invariant, then \mathbb{E}\big{[}(\lambda-X_{N}-Y_{N})^{-1}\big{]} belongs to the unital algebra generated by . Let now assume that whose eigenvectors are orthogonal to those of . This property ensures that the spikes are eigenvalues of . It implies that for any matrix in the algebra generated by , its compression is a scalar matrix. Hence in particular
[TABLE]
is a scalar matrix, and it is actually a good approximation for .
Concentration properties of unitarily invariant matrices implies that the function defined in (1.7) is close to its expectation, and so is well approximated by
[TABLE]
where denote the eigenvalues of . Hence, we retrieve the fundamental Capitaine’s relation [21] between spikes and outliers, namely in the large limit, the locations of the outliers belong to the pre-image by of the spikes.
Remark 2.2*.*
When is unitary invariant random matrix and assuming that the eigenvectors of are orthogonal to those of , the locations of outliers in depend individually on the eigenvalues of . They do not depend on its eigenvectors, and the outliers generated by a spike do not depend on the other spikes.
2.3 Random matrices with a variance profile in free probability
The asymptotic of GUE matrix with a variance profile is characterized by Shlyakhtenko in [56] in the multi-matrix setting of operator-valued free probability over the diagonal. Assuming the variance profiles are of the form \gamma_{N}^{2}(i,j)=\gamma^{2}\big{(}\frac{i}{N},\frac{j}{N}\big{)}, for some bounded real-valued function , Shlyakhtenko proved that independent GUE matrices with variance profiles are asymptotically free with amalgamation over the diagonal.
In particular, this implies that the e.s.d. of converges almost surely, and that the limit is characterized by an integral operator with kernel . The approach is in the lineage of previous works on band matrices, see [41, 17, 48], for which it was observed that to derive the limiting e.s.d. of a random matrix one can derive a system of linear equations for the diagonal of the resolvent of the matrix. An interest in Shlyakhtenko’s approach is the use of the notion of operator-valued free probability, which (in particular) nicely generalizes Pastur’s equation.
More formally, Shlyakhtenko considers in [56] the operator-valued Stieltjes transform of , which the a map between sets of diagonal matrices defined by (1.9). For any bounded function , let defined by \Lambda_{N}(i,i)=\Lambda\big{(}\frac{i}{N}\big{)}. Then, for any , the diagonal matrix , seen as a piece-wise constant function on , converges to a function in L^{\infty}\big{(}[0,1],\mathbb{C}^{-}\big{)}. The functional map is characterized the identity
[TABLE]
where for any in L^{\infty}\big{(}[0,1],\mathbb{C}\big{)},
[TABLE]
Since is linear, we say that the abstract limit of (which lives in a von Neumann algebra) is a semicircular variable with amalgamation over the diagonal. Note that the mapping defined by (1.10) is a discretization of the functional map .
Recently, the traffic method yields to the observation that asymptotic freeness over the diagonal was a generic rule for large permutation invariant random matrices with a variance profile [7]. We mention briefly a consequence of this result, referring to [45] for definitions. Let be the entry-wise product of a permutation invariant random matrix that converges in traffic distribution and of a matrix that converges in graphons topology [45, see second item in Corollary 2.19]. Let be a matrix bounded in operator norm that converges in traffic distribution. Then, under the additional assumption that is permutation invariant, and are asymptotically free over the diagonal. They converge to elements and of a von Neumann algebra endowed with a conditional expectation , and their operator-valued Stieltjes transforms satisfy
[TABLE]
where is such that the above equation is valid for . For more details about the operator-valued subordination property, we refer to [59].
A motivation of our work is to specify this statement, in a comprehensive way, when is a GUE matrix with a variance profile and to give an estimate for its operator-valued Stieltjes transform. Then, we explicit the associated deterministic equivalent and we show how to adapt Capitaine’s approach for outlier detection.
2.4 The Dyson equations and local laws in RMT
In the RMT literature, Hermitian random matrices with centered entries but non-equal distribution are referred to as generalized Wigner matrices for which many asymptotic properties are now well understood in a precise sense. For example, under the assumption that the variance profile is bi-stochastic (that is its rows and columns elements sump up to one), bulk universality at optimal spectral resolution for local spectral statistics have been established in [31] and they are shown to converge to those of the GUE. The case of a Wigner matrix with a variance profile that is not necessarily bi-stochastic has been studied in [2], and non-hermitian random matrices with a variance profile have been considered in [25, 39, 38] using the notion of deterministic equivalent.
Under mild assumptions on the variance profile , the e.s.d. of a generalized Wigner matrice converges to a limiting spectral measure for which there is generally no explicit formula. Currently, a classical method to approximate an asymptotic spectral measure is to solve a nonlinear system of deterministic equations that are referred to as the Dyson equation [2, 3, 4, 5]. For each fixed dimension , the solution of this equation yields a deterministic equivalent of the resolvent of . These equations are also equivalent to the operator-valued equation of the subordination functions where the parameter is scalar, instead of being functional. For example, the vector Dyson equation studied in details in [2] corresponds to Equation (1.12) with and a diagonal matrix. The matrix Dyson equation, introduced in [3] to study Hermitian random matrices with correlated entries and nonzero expectation, is the following nonlinear matrix equation formulated for an unknown matrix-valued function
[TABLE]
for , where is the mapping (see Equation (1.3) in [3])
[TABLE]
Thus, provided that is invertible, Equation (2.9) may be written as
[TABLE]
Hence, applying the operator of both sides of the above equality yields the fixed point Equation (1.12) with scalar parameter .
The existence and stability of the solutions of the vector and matrix Dyson equation are studied in details in [2] and [3] respectively. These deterministic vector or matrix valued functions (parametrized by ) are used to prove local laws for the resolvent of . In RMT, the derivation of local laws refers to results controlling the difference between the Stieltjes transform (or the resolvent ) and a deterministic function when is allowed to decay to zero at a rate depending on . Deriving a local law for the the Stieltjes transform is a generally a delicate problem that is more involved than proving a global law which refers to the convergence (e.g. in probability) of for a fixed value of . The notion of local semicircular law for Wigner matrices, which constituted the central open question known as Wigner-Dyson-Mehta conjecture, was solved in 2011 independently by [28] and [57]. For detailed lecture notes on this notion, we refer to [13].
In the case where and is a standard Wigner matrix satisfying mild assumptions (with a constant variance profile), then the optimal local law for the Stieltjes transform and the resolvent of reads as the following concentration inequalities [13, Theorem 2.6]: for a fixed , define the complex domain
[TABLE]
Then, denoting by the Stieltjes transform of the semicircular distribution , for any and , one has that
[TABLE]
and, uniformly for ,
[TABLE]
[TABLE]
for all and all sufficiently large . The above deviation inequalities are called optimal local laws as the rate of convergence of the error terms and as tends to zero slower than are known to be optimal for which is the support of (see e.g. [13, Section 2]). Note that when is outside , faster error rate (as ) may be obtained (see e.g. [13, Theorem 10.3]).
The derivation of optimal local laws for Hermitian random matrices with an arbitrary variance profile using the deterministic solution of a Dyson equation has been largely investigated by László Erdös and his collaborators over the last decade, and for a recent overview we refer to Erdös’s lecture notes [29]. In [2, 3], the authors make several assumptions to derive local laws, in particular they suppose that the entries of the variance profile are bounded away from below. In [30] this assumption is relaxed: a weak local law is obtained for the model without lower bound assumptions on the variance profile. The generalized Wigner matrix can possibly have correlated entries that are not necessarily Gaussian random variables. More precisely, from [30, Theorem 2.1] (see also the preliminary result [6, Lemma B.1]) the following deviation inequalities hold: for a fixed define the complex domain
[TABLE]
for some arbitrary , where is the probability measure associated to the Stieltjes transform \frac{1}{N}\mathrm{Tr}\big{[}A_{N}^{\square}(\lambda)\big{]} where is the solution of the matrix Dyson equation (2.9) (see [30, Section 2]). Then, from [30, Theorem 2.1] it follows that for any , there exists such that for any
[TABLE]
for all unit vectors , and
[TABLE]
where is a constant depending on and other quantities not depending on . Inequalities (2.11) and (2.12) hold under general assumptions on the distributions of the entries of and the additive deformation (we refer to [30, Section 2.2] for further details) but without assuming that the entries of are lower bounded. In [30] this result is referred to as a local law outside the support of as the error terms in Inequalities (2.11) and (2.12) do not decay at an optimal rate when belong to . To obtain an optimal local law when (that is in the bulk of the spectrum) a flatness condition on the variance profile is added in [30, Section 2.2] which implies that the entries of are bounded from below by positive constant. Under this supplementary condition, the following deviation inequalities hold: for fixed define the complex domain
[TABLE]
where is the density of . Then, from [30, Theorem 2.2] it follows that for any , there exists such that for any
[TABLE]
for all unit vectors , and
[TABLE]
The error terms in Inequalities (2.13) and (2.14) decay at the optimal rate when and becomes small, and this result is thus an optimal local law within the bulk of the spectrum.
Comparing Inequalities (2.11)-(2.14) with the results stated in Theorem 1.1 (when is scalar) and Corollary 1.2, it follows that we have only obtained a weak local law on the convergence of the operator-valued Stieltjes transform and the scalar Stieltjes transform to their deterministic equivalent, since the decay of the error terms in Inequalities (1.14) and (1.17) as is not optimal. Hence, when is scalar, the results of Theorem 1.1 are clearly sub-optimal as better concentration bounds already exist in the literature that also go beyond the Gaussian assumption for the entries of . Nevertheless, to the best of our knowledge, obtaining a weak local law for the operator-valued Stieltjes transform is a novel result. Hence, it would be interesting to have a generalization of optimal local laws as in [2, 3, 30] but in an operator-valued sense, that is for , when is not a scalar matrix as we consider in the current paper.
Finally, it should be mentioned that the results from Corollary 1.3 and Corollary 1.4 on the convergence of the deterministic equivalents and to the matrix-valued function are novel. Indeed, we are not aware of other works dealing with the issue of localizing outliers in the singular values distribution of the matrix when has an arbitrary variance profile. The question of the optimality of the deviation inequalities (1.19) and (1.22) is obviously left open.
3 Applications and numerical illustrations
In this section, we report results on numerical experiments on the localization of outliers in the rectangular information plus noise model (1.3) for various variance profiles and additive perturbations that may posses spikes generating outliers.
3.1 Information plus noise model with a variance profile
We introduce the model
[TABLE]
where is a low-rank matrix with spikes that are equal to the singular values of , where is diagonal matrix with positive diagonal entries and (resp. ) is the matrix whose columns are the left (resp. right) singular vectors of .
The question of locating potential outliers in model (3.1) can be answered using the approach developed in this paper for the Hermitian setting. To this end, we use the principle of Hermitian dilation [50] which corresponds to embed any rectangular matrix (with complex entries) of size within a larger Hermitian block matrix by defining
[TABLE]
Note that if one denotes by the singular values of assumed to be of rank , then the spectrum of the Hermitian matrix is
[TABLE]
where the eigenvalue [math] is of multiplicity . By applying Hermitian dilation to Equation (1.3), we obtain that
[TABLE]
which is a finite rank deformation of the GUE model . For and the entry of is a centered complex Gaussian variable with variance satisfying \frac{x_{ij}}{\sqrt{M}}=\sqrt{\frac{N+M}{M}}\big{(}\frac{x_{ij}}{\sqrt{N+M}}\big{)}. Therefore, is a GUE matrix of size with variance profile , where the zero entries of are considered as centered Gaussian variables with variance equal to zero. The additive perturbation is a matrix of rank that can be written as
[TABLE]
where
[TABLE]
is a matrix whose columns are orthonormal vectors. After introducing the Hermitian dilation of model (1.3), one may thus consider the function , analytic in each variable, that is the unique solution of the fixed point equation
[TABLE]
which holds for any . Following Equation (1.11), is the map defined on by
[TABLE]
Hence, after Hermitian dilation, one may follow the approach described in Section 1.2 to approximate of the global behavior of the empirical distribution of the singular values of and to localize potential outliers generated by the spikes of .
Remark 3.1*.*
In the RMT literature, the Hermitian dilation (3.2) of a rectangular random matrices is classically referred to as Girko’s Hermitization trick[35, 36]. As our study of a GUE matrix with variance profile allows the setting where large blocks of are equal to zero, treating the setting of the information plus noise model using Girko’s Hermitization is an immediate application of our results in the Hermitian case. As explained e.g. in [4, 5] this Hermitization trick may also be used beyond the Gaussian case. However, a direct application of the results in [2, 3] on the vector and matrix Dyson equation for the study of generalized Wigner matrices is not possible as a key assumption in these papers is that the entries of the variance profile must be bounded from below. Hence, beyond the Gaussian case, the use of Girko’s Hermitization requires a specific treatment in [4, 5] and the introduction of a system of quadratic vector equations extensively studied in [1] that relates the resolvent of to the resolvent of via the equality where denotes the upper left block of .
3.2 Additive deformation of rank one
Let us first consider the rectangular model (3.1) under the simplified setting and the low rank denoising model with an additive deformation of rank , that is
[TABLE]
where are unit vectors, is a spike, and is a rectangular Gaussian matrix with a variance profile \Gamma_{N,M}=\big{(}\gamma_{N,M}^{2}(i,j)\big{)}_{i,j}. We assume that the variance profile satisfies
[TABLE]
which ensures the normalization condition \mathbb{E}\big{[}\frac{1}{N}\mathrm{Tr}X_{N,M}X_{N,M}^{*}\big{]}=1. In all the numerical experiments we took and . Following the discussion in Section 1.2 and the principle of Hermitian dilation described in Section 3.1, a potential outlier in model (3.5) may be found by searching for a positive real such that
[TABLE]
where
[TABLE]
and is the solution of the fixed point equation (3.3), with .
Remark 3.2*.*
We choose to directly search for potential outliers by minimizing over the set of reals rather than over the set of complex values with a small and fixed value as both approaches lead to the same numerical results.
A numerical approximation (for a given value of ) is easily obtained by the following iterative procedure
[TABLE]
that is stopped for sufficiently large or when the difference between two successive iterations is sufficiently small. Since \lambda\mathbb{I}_{N+M}-\mathcal{R}_{N+M}\big{(}G_{n}^{\square}(\lambda)\big{)} is a diagonal matrix, the fixed point iteration corresponds to the numerical evaluation of the vector Dyson equation [2], and it simplifies to the vector equation
[TABLE]
In practice, to find a potential solution to equation (3.7), we use a numerical optimization procedure to obtain a minimizer of the function \lambda\mapsto\det\big{(}\beta_{2}^{\square}(\lambda)\big{)} over . To this end, we have used Python’s command fminsearch which is based on the Nelder-Mead simplex method. Then, if the value \det\big{(}\beta_{2}^{\square}(\bar{\lambda}(\theta))\big{)} is sufficiently close to zero, we conclude that is an outlier.
Finally, a smooth approximation of the singular values distribution (s.v.d.) of at location may be obtained from the inverse Stieltjes transform formula (1.5). Letting , we define for small enough. Now, recall that is an approximation of the dilation matrix ) whose eigenvalue values are [math] (with multiplicity ) and where are the singular values of , and that the inverse Stieltjes transform (1.5) amounts to approximate a measure by a convolution with the Cauchy kernel . Therefore, an approximation of the s.v.d. of is given by the density
[TABLE]
3.2.1 Constant variance profile
We propose to validate this way of localizing outliers by first considering the standard case where the variance profile has constant entries equal to one. This setting corresponds the so-called Gaussian spike population model for which the asymptotic behavior (as ) of the singular values of is well understood [14, 27, 55, 44] when the rank of the additive deformation is held fixed. In the asymptotic framework where the sequence is such that with , it is well known [9] that the empirical distribution of the singular values of converges, as , to the quarter circle distribution if , or to its generalized version if , called the Marchenko-Pastur distribution, which is supported on the compact interval with where is the so-called bulk (right) edge. Now, if one denotes by (recall that ) the singular values of , then the following result holds (see e.g. Theorem 2.8 in [14] and Proposition 9 in [55]). Almost surely, one has that
[TABLE]
and where
[TABLE]
The interpretation of this result, called the BBP transition after [10], is as follows. If the spike in model (3.5) is larger than then an outlier exists and it is asymptotically located at . To the contrary, if then there exists no outlier as the largest singular value converges to the bulk edge .
For a constant variance profile, an explicit solution of the equation \det\big{(}\beta_{2}^{\square}(\lambda)\big{)}=0 exists as stated below.
Lemma 3.3**.**
Assume that for any . Then, the Equation (3.7) admits a solution given by
[TABLE]
Proof.
Let us first determine the solution of the fixed point equation (3.3) for a constant variance profile, namely for any . Using the particular structure of the variance profile and the expression (3.4) of , one obtains by simple calculations that G_{\mathcal{D}(H)}^{\square}(\lambda\mathbb{I}_{N+M})=\left[\begin{array}[]{cc}g_{N}^{\square}(\lambda)\mathbb{I}_{N}&0\\ 0&g_{M}^{\square}(\lambda)\mathbb{I}_{M}\end{array}\right], where are complex-valued functions satisfying and
[TABLE]
Inserting this expression for into (3.8), one obtains that
[TABLE]
Then, by simple calculations, it can be shown that the equation \det\big{(}\beta_{2}^{\square}(\lambda)\big{)}=0 admits a solution given by (3.13) provided that . ∎
Note that the condition guarantees that . Hence, we retrieve the expression (3.12) of the asymptotic location of an outlier in the Gaussian spike population model where the asymptotic ratio is replaced with its non-asymptotic approximation . As expected, we also remark that the localization of an outlier does not depend on the singular vectors and of the additive perturbation .
In Figure 1(a), we display the histogram of the singular values of one realization of with , where and are chosen to be unit vectors with constant entries. There is clearly an outlier outside the interval . In Figure 1(a), we also plot the curve which shows that the density defined by (3.11) is a very satisfactory approximation the distribution of the singular values of . In Figure 1(b), we plot the curve for , which gives the location of outliers for any .
For a set of regularly spaced values of on , we also report the results of the numerical procedure that we use to compute a minimizer of the function \lambda\mapsto\det\big{(}\beta_{2}^{\square}(\lambda)\big{)} over . In Figure 1(c), we display the curve \theta\mapsto\det\big{(}\beta_{2}^{\square}(\bar{\lambda}(\theta))\big{)}. It can be seen that this curve is close to zero for , and that it does not vanish for smallest values of which is in agreement with the fact that there is no outlier for . In Figure 1(b), we also display the curve found by numerical minimization which coincides with for . Interestingly, for all values of smaller than it appears that , suggesting that \lambda\mapsto\det\big{(}\beta_{2}^{\square}(\lambda)) admits a minimizer at the bulk edge.
3.2.2 Piecewise constant variance profile
We now consider the following example of a piecewise constant variance profile
[TABLE]
where denotes the vector of length with all entries equal to one, and are positive constant such that . Then, we compare two settings where either and are unit vectors with constant entries, or (resp. ) is equal to the first vector (resp. ) of the canonical basis of (resp. ).
In Figure 2, we display the histogram of the singular values of one realization of for different values of for and . When and are unit vectors with constant entries, a spike clearly generates an outlier at , while in the case and there is no outlier for such values of the spike. For each setting, we also display in Figure 3 the curves and \theta\mapsto\det\big{(}\beta_{2}^{\square}(\bar{\lambda}(\theta))\big{)} where is found by numerical minimization of \lambda\mapsto\det\big{(}\beta_{2}^{\square}(\lambda)\big{)} over . In the case where and have constant entries, the value of \det\big{(}\beta_{2}^{\square}(\bar{\lambda}(\theta))\big{)} is close to zero when which confirms the existence of outliers for values of the spike within this interval. When and , one has clearly that \det\big{(}\beta_{2}^{\square}(\bar{\lambda}(\theta))\big{)}\neq 0 when and thus, for such spikes, there is no outlier. In both settings, when is sufficiently large (e.g. ), there exists an outlier but with a location depending on the values of and .
3.2.3 Bernoulli variance profile
We now consider variance profiles whose entries may be equal to zero and are chosen randomly (and independently) as follows. Each entry of takes either the value zero with probability (for some ) or a fixed positive value with probability . After randomly fixing the entries of in this way, the value of is chosen such that the normalisation condition (3.6) is satisfied. In Figure 4, we display the histogram of the singular values of one realization of for , with , and for various Bernoulli variance profiles by letting the sampling probability ranging from to . For values of larger than , the value is an accurate approximation of the location of an outlier in the singular values distribution of . The shape of the smooth approximation by of the singular values distribution of clearly depends on the value of .
3.2.4 Doubly stochastic variance profile
We finally assume that , and we consider the setting where is a doubly stochastic matrix. Under such an assumption, an explicit solution of the equation \det\big{(}\beta_{2}^{\square}(\lambda)\big{)}=0 exists as stated below.
Lemma 3.4**.**
Assume that is a doubly stochastic matrix. Then, the Equation (3.7) admits a solution given by
[TABLE]
Proof.
Under the assumption that the variance profile is doubly stochastic, it can be easily shown that the solution to (3.3) is a scalar matrix
[TABLE]
where is complex-valued function satisfying for . Therefore, one has that which is the Stieltjes transform of the semicircular law. Hence, using exactly the same calculations than those made for a constant variance profile to derive Lemma 3.3, one obtains that, for any unit vectors and , the equation
[TABLE]
admits a solution given by (3.15) provided that . ∎
Interestingly, for a doubly stochastic variance profile, we obtain exactly the expression (3.12) of the asymptotic location of an outlier in the standard Gaussian spike population model for the ratio . In Figure 5, we display the histogram of the singular values of one realization of with and and chosen to be unit vectors with constant entries. The normalized variance profile of is chosen as follows
[TABLE]
where are permutation matrices (obtained by random permutations of the columns of the identity matrix ). For small values of , such variance profiles have many entries equal to zero. In Figure 5, we display the histogram of the singular values of one realization of with for different values of . For , there is clearly an outlier located approximately at . The quality of the smooth approximation by of the singular values distribution of also clearly depends on the value of (maximum value of the entries of the variance profile ) which is consistent with our theoretical results in Theorem 1.1 on the control of the deviation between the deterministic equivalent and the Stieltjes transform .
3.3 General deformed models
Let us now consider the general setting of the rectangular information plus noise model (3.1) with .
3.3.1 Simulated model
Taking again and we generate one realization from the model as follows. The matrix is a Gaussian matrix with piecewise constant variance profile given by (3.14). Then, we generate a matrix with i.i.d. real entries sampled from a Gaussian distribution with zero mean and variance , that we write using singular value decomposition (SVD) as . Denoting (resp. ) the left (resp. right) singular vectors associated to the three largest singular values of , we finally define
[TABLE]
The histograms of the s.v.d. of , and are displayed in Figure 6. It can be seen that the spikes of clearly generate 3 outliers.
In Figure 6(c), we also display the smooth approximation of the s.v.d. of the random matrix using the inverse Stieltjes transform (1.5) of . Since is not a diagonal matrix, the deterministic equivalent of the operator-valued Stieltjes transform of is obtained by iterating the following matrix equation
[TABLE]
In Figure 6(d), we also report the curves of the mapping , where is the deterministic equivalent defined by (1.18), and of the mapping defined by (1.21). As is non diagonal, the second deterministic equivalent gives the right prediction of the locations of the outliers which is not the case for the first one . This is confirmed by the numerical simulations reported in Figure 6(c), where it can be seen that the zeros of the mapping correspond to the locations of the true outliers in the s.v.d. of .
3.3.2 A model of noisy images with heteroscedasticity
To conclude this section on numerical experiments, we study an example inspired by the problem of low-rank matrix denoising in image processing in the presence of Poisson noise. In this setting, one observes a data matrix such that each -th entry is independently sampled from a Poisson distribution with parameter . Under such an assumption, the expectation and variance of each entry are thus equal to .
Therefore, the following rectangular information plus noise model
[TABLE]
where is a rectangular Gaussian matrix with a variance profile \Gamma_{N,M}=\big{(}\gamma_{N,M}^{2}(i,j)\big{)}_{i,j}, may be viewed as a prototype for studying low-rank matrix denoising in the presence of Poisson noise (with ), as considered e.g. in [62]. We shall refer to model (3.18) as the Gaussian setting with equal mean and variance. In Figure 7, we display the image made of the entries of the normalized variance profile that is considered in these numerical experiments, as well as the histogram of the singular values of this matrix which is scaled so that it satisfies the normalization condition:
[TABLE]
Now, let us consider the SVD of the normalized variance profile . For any , model (3.18) can be written as
[TABLE]
where is diagonal matrix whose elements are the largest singular values of and (resp. ) is the matrix made of the associated left (resp. right) singular vectors, and
[TABLE]
is the matrix obtained by keeping only the remaining smallest singular values in the SVD of the normalized variance profile.
In Figure 8, we display the histogram of the s.v.d. of sampled from model (3.18). We also report the smooth approximation of the singular value distributions of the random matrices and for using the inverse Stieltjes transform (1.5) of . As described previously, for the matrix , the deterministic equivalent of its operator-valued Stieltjes transform is obtained by iterating the vector Dyson equation (3.10). For the matrix , such a deterministic equivalent is obtained by iterating the matrix equation (3.17).
In Figure 8, we also report, for , the values of the mapping \lambda\mapsto\det\big{(}\beta_{2k}^{\square}(\lambda)\big{)} and \lambda\mapsto\det\big{(}\tilde{\beta}^{\square}_{2k}(\lambda)\big{)} for . Again, this illustrates the benefits of using instead of for outliers detection. Moreover, for all one predicts accurately outliers using the mapping \lambda\mapsto\det\big{(}\tilde{\beta}^{\square}_{2k}(\lambda)\big{)}.
4 Organization of the proofs
We first describe the mains steps to derive the proof of Theorem 1.1 using the notation of Section 1.2. In Section 5, following the method and terminology of Haagerup and Thorbjørnsen in [37], we start by proving a Master equality (see Lemma 5.7) involving the expectation of the generalized resolvent that can be decomposed as follows
[TABLE]
with
[TABLE]
and F_{N}(\Lambda)=\big{(}\Omega_{H_{N}}(\Lambda)-Y_{N}\big{)}^{-1}E_{N}, where is the matrix of covariance between and \mathcal{R}_{N}\big{(}\mathbb{E}[G_{X_{N}+Y_{N}}(\Lambda)]\big{)} given explicitly in (5.14). We deduce this result from an identity on the resolvent that is a consequence of the Gaussian integration by part formula (see Lemma 5.5 and Lemma 5.6 below).
Following the heuristic of Section 2.2, in particular (2.5), the matrix is a good candidate to approximate an operator-valued subordination function. Applying the operator on both sides of equality (4.1) provides the following approximate equation for operator-valued Stieltjes transforms:
[TABLE]
Equation (4.2) tells that the expectation of satisfies, up to additive error term, the fixed point equation (1.12) that defines its deterministic equivalent. Then, using Gaussian Poincaré inequality (see Lemma 5.9), we obtain an upper bound on and that we call Master inequalities (see Lemma 5.8 below), following again the terminology of Haagerup and Thorbjørnsen [37]. It will be finally shown that
[TABLE]
and if is diagonal then
[TABLE]
In Section 6 we prove the existence of the deterministic equivalent , solution of the equation as in (4.2) taken for . We also prove that the upper bound (4.3) for \big{\|}\Delta[F_{N}(\Lambda)]\big{\|} implies a bound for the difference \big{\|}\mathbb{E}\big{[}G_{H_{N}}(\Lambda)\big{]}-G^{\square}_{H_{N}}(\Lambda)\big{\|}, thanks to an analysis of regularity of the fixed point problem (1.12). More precisely, is a good approximation of out of a small strip around the real axe as stated below.
Lemma 4.1**.**
For all and all such that we have
[TABLE]
where is the bound in the r.h.s. of (4.3). Moreover, if is diagonal and , then the above estimate holds with as in the r.h.s. of (4.4).
The rest of the proof of Theorem 1.1 is finally based on the control of the difference between the generalized resolvent and the expectation of . As explained in Section 7, the comparison between the random quantity and the deterministic equivalent follows from the combination of Lemma 4.1 and Gaussian concentration inequality of Lipschitz functions allowing to show that the entries of the generalized resolvent are close to their expectation with high probability. Finally, Section 7 ends with a mathematical justification of the convergence of the numerical method used to approximate the solution of the fixed point equation (1.12).
5 The approximate subordination property
5.1 Notation and preliminaries
We let be the set of integers between and . Then, we recall some basic properties of matrices with complex entries that we repeatedly use in the proof. For any matrix in we denote by its operator norm, namely
[TABLE]
we stands for the standard scalar product on . We recall if is a Hermitian matrix then is the spectral radius of , and in general it is the square-root of the spectral radius of . In particular it satisfies the -norm condition . We denote by and the real and imaginary parts of , which are Hermitian matrices defined by
[TABLE]
We write (resp. ) whenever the matrix is Hermitian and positive (resp. positive definite), and (resp. ) if satisfies this property.
Lemma 5.1**.**
Let in such that . Then is invertible and
[TABLE]
Proof.
We follow the proof of Lemma 3.1 in [37]. For any unit vector in , since and are real, we have
[TABLE]
Hence is injective, so it is invertible. Since , we get from the previous lower bound that . ∎
For a diagonal matrix , note that and whenever for any . Recall that we defined \Delta(A)=\underset{i\in[N]}{\mathrm{diag}}\big{(}A(i,i)\big{)} for any matrix .
Lemma 5.2**.**
For any in , one has
[TABLE]
Moreover, implies and implies .
Proof.
We have \|\Delta(A)\big{\|}=\underset{i\in[N]}{\max}\big{|}A(i,i)\big{|}=\underset{i\in[N]}{\max}\big{|}\langle e_{i},Ae_{i}\rangle\big{|}\leqslant\|A\|, where denotes the canonical basis of . Moreover, if then for some Hermitian matrix . Hence for any , \Delta(A)(i,i)=-\sum_{j=1}^{N}\big{|}B(i,j)\big{|}^{2}\leqslant 0, and so we get . If moreover then is invertible so the -th line of is nonzero for any and hence . ∎
If is an invertible diagonal matrix, note also that \|\Lambda^{-1}\|=\Big{(}\underset{i\in[N]}{\min}|\Lambda(i,i)|\ \Big{)}^{-1}. We use several times the following direct consequence of Lemmas 5.1 and 5.2.
Lemma 5.3**.**
For any Hermitian matrix and any diagonal matrix such that , the matrix is invertible and
[TABLE]
Hence the map G_{A}:\Lambda\mapsto\Delta\big{[}(\Lambda-A)^{-1}\big{]} is well defined on , and it satisfies \big{\|}G_{A}(\Lambda)\big{\|}\leqslant\|(\Im m\,\Lambda)^{-1}\| and for all .
Recall that we defined \mathcal{R}_{N}(G)=\underset{i\in[N]}{\mathrm{diag}}\Big{(}\sum_{j=1}^{N}\frac{\gamma^{2}_{i,j}}{N}G_{j,j}\Big{)} for any diagonal matrix . Throughout the paper, we denote .
Lemma 5.4**.**
For any diagonal matrix such that , one has . Moreover, for any diagonal matrix , the following operator norm bound holds
[TABLE]
Proof.
For any such that , since the matrix has nonnegative entries, we have
[TABLE]
where denotes the matrix will entries equal to zero except the -th one which is equal to one. Hence the diagonal entries of are indeed nonpositive. Moreover, for any diagonal matrix we have
[TABLE]
Since , we obtain the expected inequality. ∎
5.2 The Master equality
The notation are as in Section 1.2, in particular we denote . We start with the following observation.
Lemma 5.5**.**
For any , diagonal matrix with positive imaginary part, recalling that G_{H_{N}}(\Lambda)=\Delta\big{[}(\Lambda-H_{N})^{-1}\big{]}, we have the equality between the matrices
[TABLE]
Lemma 5.5 is a consequence of the well-known Gaussian integration by part formula (see e.g. Lemma 3.3 in [37]) that we recall below.
Lemma 5.6**.**
Let be a continuously differentiable function, and a sequence of independent centered real Gaussian variables with possibly different variances for . Then, under the conditions that and its first order derivatives are polynomially bounded, one has that
[TABLE]
Proof of Lemma 5.5.
Let us write the random matrix in an appropriate orthonormal basis to make its dependency on only real Gaussian variables more explicit. Let be the canonical basis of matrices, and define
[TABLE]
Then, can be decomposed as
[TABLE]
where the are i.i.d. real Gaussian random variables, centered and such that has variance . This implies that
[TABLE]
By the Gaussian integration by part (5.5), we have
[TABLE]
Now, recalling that , we can compute
[TABLE]
for any direction , and get the following relation
[TABLE]
Note that (5.3) ensures that the function and its derivatives are bounded, so we can correctly apply (5.5). Moreover, since , we have for any matrix
[TABLE]
Combined with (5.12), the equality implies the expected result
[TABLE]
∎
Let us now introduce defined as the covariance between the matrices \mathcal{R}_{N}\big{(}G_{H_{N}}(\Lambda)\big{)} and , namely
[TABLE]
We can now state and prove the so-called Master equality introduced in Equation (4.1).
Lemma 5.7** (Master equality).**
For any , we define the diagonal matrix
[TABLE]
Then, we have , and the following equality holds for all :
[TABLE]
Proof.
Starting with the left hand side in (5.4), we want to obtain an expression involving only generalized resolvent. Recall that . If we where solely considering the random matrix and assume , we should write
[TABLE]
When is non zero, we first fix a deterministic matrix whose choice is determined later on, and multiplying on the left by our expression under consideration: we have
[TABLE]
Moreover, since is deterministic, the master equality (5.4) is equivalent to
[TABLE]
Introducing the map f:A\mapsto\mathbb{E}\big{[}(\Omega-Y_{N})^{-1}A(\Lambda-H_{N})^{-1}\big{]} for any random matrix , we obtain
[TABLE]
Since is linear, we have
[TABLE]
We set \Omega=\Omega_{H_{N}}(\Lambda)=\Lambda-\mathcal{R}_{N}\big{(}\mathbb{E}\big{[}G_{H_{N}}(\Lambda)\big{]}\big{)}, so that the first term in the above equality vanishes. By Lemma 5.4, we have that \Im m\big{(}\mathcal{R}_{N}\big{(}G_{H_{N}}(\Lambda)\big{)}\big{)}\leqslant 0, so belongs to and satisfies . Hence we obtain the following expression
[TABLE]
Since is deterministic, the above identity completes the proof of Lemma 5.7. ∎
5.3 The Master inequality
5.3.1 Statement and use of Poincaré inequality
We prove the following estimates on defined in (5.14) as the covariance between the matrices \mathcal{R}_{N}\big{(}G_{H_{N}}(\Lambda)\big{)} and
Lemma 5.8** (Master inequality).**
Recall that we denote . For any belonging to , we have
[TABLE]
By Lemma 5.7 one has that . Hence as announced in the presentation of the proof of Section 4, Lemma 5.8 implies that the operator-valued subordination property (4.2) approximatively holds up to an error term \Delta[F_{N}(\Lambda)]=\Delta\big{[}(\Omega_{H_{N}}-Y_{N})^{-1}E_{N}\big{]} satisfying
[TABLE]
Moreover, if is diagonal then so is and we have
[TABLE]
The rest of the section is devoted to the proof of Lemma 5.8. To abbreviate the notation, we define the random matrices (and functions of the diagonal parameter )
[TABLE]
so that E_{N}=\mathbb{E}\big{[}\overset{\circ}{D}_{N}\overset{\circ}{A}_{N}\big{]}. We first use the Cauchy-Schwarz inequality for the norm
[TABLE]
We recall the proof of (5.20) from [40, Lemma 3.1 (a1)]. By the singular value decomposition, is the supremum of \big{|}\big{\langle}\mathbb{E}[\overset{\circ}{D}_{N}\overset{\circ}{A}_{N}]x,y\big{\rangle}\big{|} over all unit vectors and in . Moreover, we have
[TABLE]
Hence the inequality (5.20). We hence deduce a first estimate
[TABLE]
For the diagonal of we shall use the classical Cauchy-Schwarz inequality:
[TABLE]
We now state the Gaussian Poincaré inequality (see e.g. Proposition 4.1 in [37]), which allows use to estimate the variances \mathbb{V}\textrm{ar}\big{(}A_{N}(k,k)\big{)} and \mathbb{V}\textrm{ar}\big{(}D_{N}(k,k)\big{)} that appear in Inequalities (5.21) and (5.22).
Proposition 5.9** (Gaussian Poincaré inequality).**
Let be a continuously differentiable function, and a sequence of independent centered real Gaussian variables with possibly different variances for . Then, under the condition that and its first order derivatives are polynomially bounded, one has that
[TABLE]
where , is the gradient of , and is the standard Euclidean norm of a vector with complex entries.
We write the matrices and as functions of the independent real Gaussian random variables defined in (5.10), namely: recalling , we define for any real matrix the diagonal matrices
[TABLE]
where is the basis of Hermitian matrices defined by (5.9), so that we have \Delta[A_{N}]=f_{1}\big{(}(x^{\prime}_{i,j})_{i,j}\big{)} and D_{N}=f_{2}\big{(}(x^{\prime}_{i,j})_{i,j}\big{)} for the coordinates of in the basis , see (5.10). By the same computation as for the derivative in (5.11), and by the linearity of the maps and , we obtain
[TABLE]
For any , we apply Proposition 5.9 (Gaussian Poincaré inequality) to the -th diagonal entry of and of , and hence get
[TABLE]
5.3.2 Estimation of the terms given by Poincaré inequality
Hence to obtain the required estimates for (5.21) and (5.22), one should find upper bounds for (5.23) and (5.24). This is the purpose of this section, where we derive below the two estimates (5.26) and (5.28), which allow to complete the proof of Lemma 5.8.
We first write in a different way the term \sum_{i,j}\big{|}\big{[}A_{N}F_{i,j}A_{N}\big{]}(\ell,\ell)\big{|}^{2}. Recall that denotes the canonical basis of . Recalling that the entry of a matrix is equal to , that the elementary matrix equals . Since is a Hermitian matrix, we have
[TABLE]
Note also that (5.13) implies the following equality, valid for any matrix :
[TABLE]
Using (5.25) for , we get
[TABLE]
Hence the application of Poincaré inequality (5.23) yields
[TABLE]
Similarly, we re-write the term \sum_{i,j}\big{|}\mathcal{R}_{N}\big{(}\Delta\big{[}A_{N}F_{i,j}A_{N}\big{]}\big{)}(\ell,\ell)\big{|}^{2}. Since for any matrix ,
[TABLE]
we obtain
[TABLE]
Using (5.25) for , we get the expressions and estimates
[TABLE]
The Cauchy-Schwarz inequality for implies
[TABLE]
Hence with (5.24), the above inequality gives
[TABLE]
Unfortunately, the Poincaré inequality does not conclude to an interesting estimate for \big{\|}\mathbb{E}\big{[}\overset{\circ}{A}_{N}^{*}\overset{\circ}{A}_{N}\big{]}\big{\|} that will be roughly bounded by . Inserting the estimates (5.28) in (5.21) give
[TABLE]
Moreover, (5.26) and (5.21) implies
[TABLE]
which proves Inequality (5.17) and complete the proof of Lemma 5.8.
6 The fixed point equation
In this section, we prove the existence and uniqueness of the deterministic equivalent , and we derive an estimate for the difference G_{H_{N}}^{\square}-\mathbb{E}\big{[}G_{H_{N}}\big{]}.
6.1 Fixed point formulation
Recall that for any in , we denote G_{Y_{N}}(\Lambda)=\Delta\big{[}(\Lambda-Y_{N})^{-1}\big{]}. For any in we consider the function
[TABLE]
So is solution of Equation (1.12) of Theorem 1.1 if and only if is a fixed point of for any . Note that by Lemma 5.4, when then and so . Hence we can correctly evaluate the function in this diagonal matrix and the expression defining makes sense. Moreover, by the last statement of Lemma 5.3 we indeed have . We shall use the next statement later.
Lemma 6.1**.**
For any the function is bounded and Lipschitz for the operator norm: for any ,
[TABLE]
Proof.
[TABLE]
Moreover, by Lemma 5.2 and the fact that is an algebra norm, for any in we have
[TABLE]
By (5.3) we have \big{\|}\big{(}\Lambda-\mathcal{R}_{N}(G)-Y_{N}\big{)}^{-1}\big{\|}\leqslant\big{\|}\Im m\big{(}\Lambda-\mathcal{R}_{N}(G)\big{)}^{-1}\big{\|}. But by Lemma 5.4 we have \Im m\big{(}\Lambda-\mathcal{R}_{N}(G)\big{)}\geqslant\Im m\,\Lambda and so \big{\|}\big{(}\Lambda-\mathcal{R}_{N}(G)-Y_{N}\big{)}^{-1}\big{\|}\leqslant\|(\Im m\,\Lambda)^{-1}\|. The same inequality holds for instead of . These inequalities together with the estimate \big{\|}\mathcal{R}_{N}(G-G^{\prime})\big{\|}\leqslant\gamma^{2}_{\mathrm{max}}\|G-G^{\prime}\| of Lemma 5.4 yield
[TABLE]
∎
By Banach fixed-point theorem and Lemma 6.1, we get the existence and uniqueness a priori of the deterministic equivalent on a region on .
Corollary 6.2**.**
For any such that there exists a unique deterministic diagonal matrix such that G_{H_{N}}^{\square}(\Lambda)=\psi_{\Lambda}\big{(}G_{H_{N}}^{\square}(\Lambda)\big{)}.
6.2 The deterministic equivalent
6.2.1 Setting of the problem
We want to extend the previous corollary for any in . The difficulty is that is not contractive in general. Fortunately, it will be enough in our problem to consider uniqueness in the class of analytic function in several variables. Recall from [54] that for any open set of , we say that a function is analytic on whenever for any the function
[TABLE]
are analytic in each variable .
Lemma 6.3**.**
There exists a unique deterministic analytic map such that G_{H_{N}}^{\square}(\Lambda)=\psi_{\Lambda}\big{(}G_{H_{N}}^{\square}(\Lambda)\big{)} for any . Moreover, for any ,
[TABLE]
The remainder of the subsection is devoted to the proof of Lemma 6.3. By [54, Conclusion 1.2.1.2], the analytic continuation principle holds for analytic maps in several variables. Hence, by Corollary 6.2 we know that there exists at most one analytic map solution of the fixed point problem on , since all solutions must coincide in . In Section 6.2.2 we prove the existence of such function and that it satisfies the estimate (6.2). Then, Section 6.2.3 will be dedicated to the proof of analyticity.
6.2.2 Large random matrix model
Given , and fixed we consider an intermediate sequence of Hermitian random matrices of size by . Seeing a generic element of as an element of , we denote
[TABLE]
Then, we consider two new random and deterministic matrices.
-
Let be a G.U.E. matrix with variance profile , where is the matrix whose all entries are one. Hence the variance profile is constant on blocks of size and we can write
[TABLE]
where the are complex Gaussian random variables, i.i.d. up to the Hermitian symmetry, centered and such that has variance .
-
We denote by , where is the identity matrix. It is a deterministic matrix diagonal by blocks of size , so that .
We set . To avoid ambiguity, we denote by a generic element of and a generic element of . We consider the diagonal of the generalized resolvent
[TABLE]
and the deterministic function
[TABLE]
Note first that since we know that up to a subsequence has a limit as goes to infinity for any . Moreover, the same computation as Lemma 6.1 yields
[TABLE]
Letting going to infinity along a subsequence, this implies that the estimate (6.2) is valid for any accumulation point of .
We shall now prove that converges when goes to infinity to a solution of the fixed point problem. Thanks to Lemma 5.8, we may now apply Equality (4.2) to the random matrix : for any , we have
[TABLE]
where
[TABLE]
for any , and we have the estimates and if is diagonal. Note that is indeed the maximum of the variances in the profile . Moreover, because of the definition of the map is given by: for any , for any
[TABLE]
Since the above expression does not depends on , this proves that
[TABLE]
Moreover note that if then . Hence we get for any
[TABLE]
Hence, applying in (6.6) yields the following formula:
[TABLE]
Since goes to zero as goes to infinity and by the continuity of , letting go to infinity proves that any accumulation point of is solution of the fixed point problem that satisfies (6.2) thanks to Inequality (6.5).
6.2.3 Analiticity
Let us justify that the quantities under consideration up to now are analytic functions.
Lemma 6.4**.**
For any Hermitian random matrix , the function \Lambda\mapsto\mathbb{E}\big{[}(\Lambda-M)^{-1}\big{]} is analytic on .
Proof.
Let us first assume that the matrices are deterministic. We prove the lemma by induction of the size of the matrices. Since for any the map is analytic on , the lemma is true for . From now we fix and we assume the lemma is true for all deterministic Hermitian matrices of size .
We write and recall that we denote . For any , let be the inverse of the by matrix obtained from by removing the -th line and column. Let be the vector of size obtained from the -th column of by removing the -th entry. Recall the Schur complement formula [8, Appendix A.1.4]:
[TABLE]
By induction hypothesis, is analytic on . By Lemma 5.3 we have
[TABLE]
Hence the maps are analytic in each variable for each , and hence so are the maps for any .
Let now assume that is random. Each realization is analytic and the map is bounded. Hence \Lambda\mapsto\mathbb{E}\big{[}(\Lambda-M)^{-1}\big{]} is also analytic. ∎
Hence the map defined by (6.4) is indeed analytic. Since it is Lipschitz by Inequality (6.5), it follows that every accumulation point of the sequence is also analytic. This finishes the proof of Lemma 6.3.
6.3 Stability of the fixed point equation and proof of Lemma 4.1
Let be the deterministic equivalent, unique analytic solution of the fixed point problem
[TABLE]
For reading convenience, we recall that \mathbb{E}\big{[}G_{H_{N}}(\Lambda)\big{]}=\mathbb{E}\Big{[}\Delta\big{[}(\Lambda-H_{N})^{-1}\big{]}\Big{]} satisfies the approximate subordination property, namely
[TABLE]
where \Theta_{N}(\Lambda)=\Delta\big{[}(\Omega_{H_{N}}(\Lambda)-Y_{N})^{-1}E_{N}(\Lambda)\big{]}. The operator norm of satisfies by Lemma 5.8 where
[TABLE]
in general, and
[TABLE]
if is diagonal. The purpose of this section is to prove Lemma 4.1, giving an estimate for the norm of the difference \big{\|}G_{H_{N}}^{\square}(\Lambda)-\mathbb{E}\big{[}G_{H_{N}}(\Lambda)\big{]}\big{\|}.
We define two diagonal matrices by
[TABLE]
Provided we can justify that belongs to , we have
[TABLE]
With denoting the identity matrix and using Lemma 5.4 and the bound (6.8) for , we have
[TABLE]
where is as in (6.8), or as in (6.9) when is diagonal. Assuming that
[TABLE]
we indeed have and so is solution of the fixed point problem for . If is diagonal, the same conclusion holds whenever
[TABLE]
Hence, by Lemma 6.3 we obtain the equality and so, by Equalities (6.7) and (6.10), we obtain
[TABLE]
By Lemma 6.4, the map is \Lambda\mapsto\mathbb{E}\big{[}G_{H_{N}}(\Lambda)\big{]} is analytic on . Recall that \Theta_{N}(\Lambda)=\Delta\big{[}(\Omega_{H_{N}}(\Lambda)-Y_{N})^{-1}E_{N}\big{]} where is defined in (5.14) and is defined in (5.15). One checks easily that the map is analytic on , which implies that so are and . Hence Equality (6.14) extends by analyticity for all such that and we get
[TABLE]
Moreover, with the same proof as for (6.2), we have the estimate
[TABLE]
We have by Lemma 5.4, Moreover, under the assumption
[TABLE]
we obtain by (6.11) that
[TABLE]
Hence provided that Condition (6.17) holds, we have . Combining (6.15) with (6.16) and (6.8), the previous estimates on and give
[TABLE]
which completes the proof of Lemma 4.1.
7 Analysis of the resolvent
We now have all the ingredients to control the difference between the resolvent and the deterministic equivalent which will finally complete the proof of Theorem 1.1.
7.1 Expectation out of the diagonal
We first establish results allowing to show that the expectation of the resolvent is a diagonal matrix. Recall that a random matrix is unitarily invariant whenever has the same law as for any unitary matrix .
Lemma 7.1**.**
Let be a by unitarily invariant random matrix whose entries have finite moment of any orders and let Then for any , the matrix \mathbb{E}\big{[}(\Sigma\circ A)^{n}\big{]} is diagonal.
Proof.
For any we have
[TABLE]
where denotes the -th entry of the matrix .
We shall prove that for any then \mathbb{E}\Big{[}\prod_{k=1}^{n}A(i_{k},i_{k+1})\Big{]}=0 when . For this purpose we introduce a matrix function that depends on an implicit parameter : for any anti-Hermitian matrix , i.e. such that , and for any , we denote . Note that and the derivative of with respect to is . Moreover, the unitary invariance of implies that and have the same law. In particular for any and any we have
[TABLE]
We now differentiate the above equality with respect to and take : using Leibniz formula,
[TABLE]
Recall that the above equality is a priori valid under the assumption that is anti-Hermitian. But the relation is linear in , and the set of anti-Hermitian matrices spans as a vector space (any matrix can be written as a linear combination of Hermitian matrices and a matrix is Hermitian whenever is anti-Hermitian). We can hence specify the equality for the elementary matrix . Note that for any , we have
[TABLE]
Hence we obtain from Equation (7.1) a telescopic sum
[TABLE]
If then we get \mathbb{E}\big{[}\prod_{k=1}^{n}A(i_{k},i_{k+1})\big{]}=0 for any and hence by Equation (7.1) we deduce that \mathbb{E}\big{[}(\Sigma\circ A)^{n}\big{]} is a diagonal matrix.
∎
Corollary 7.2**.**
Let and be as in Lemma 7.1. Assume moreover that is Hermitian. Then for any , the expectation of the generalized resolvent \mathbb{E}\big{[}(\Lambda-\Sigma\circ A)^{-1}\big{]} is a diagonal matrix.
Proof.
We first assume that there is a constant such that almost surely one has . For any matrix we denote by its Frobenius norm \big{\|}M\big{\|}_{F}=\big{(}\mathrm{Tr}\big{[}M^{*}M\big{]}\big{)}^{\frac{1}{2}} and we recall that . Hence we have for ,
[TABLE]
where denotes the -th entry of the matrix , and . For any such that , we have that , and thus the following identity holds
[TABLE]
where the convergence of the sum is normal. In particular we can interchange summation and expectation, namely
[TABLE]
Moreover, we have the equality where . Hence by Lemma 7.1 for any and any the matrix \mathbb{E}\big{[}\big{(}(\Sigma\circ A)\Lambda^{-1}\big{)}^{n}\big{]} is diagonal. So for any such that , the matrix \mathbb{E}\big{[}G_{X_{N}}(\Lambda)\big{]} is also diagonal. This fact extends for any by analytic continuation thanks to Lemma 6.4.
To treat the general case, we use a classical spectral truncation argument. Let us denote by and the eigenvalues and the associated eigenvectors of , so that we have . For any , we denote by the matrix A^{(B)}=\sum_{i=1}^{N}\lambda_{j}\mathbbm{1}\big{(}|\lambda_{j}|\leqslant B\big{)}u_{i}^{*}u_{i}, where denote the indicator function. Hence is uniformly bounded in operator norm by . By the previous case, the matrix \mathbb{E}\big{[}(\Lambda-\Sigma\circ A^{(B)})^{-1}\big{]} is diagonal. Moreover, we have
[TABLE]
With fixed, we get that almost surely as tends to infinity the matrix converges to . Since the matrices are bounded in operator norm, the convergence holds in expectation. In particular, \mathbb{E}\big{[}(\Lambda-\Sigma\circ A)^{-1}\big{]} is the limit of a diagonal matrix so it is diagonal. ∎
7.2 Concentration argument and proof of the main results
We now complete the proof of Theorem 1.1 by combining Lemma 4.1 with a concentration argument for the resolvent towards its expectation \mathbb{E}\big{[}(\Lambda-H_{N})^{-1}\big{]}.
Lemma 7.3**.**
Let . Then, for any pair of unit vectors (that is ), one has that, for all ,
[TABLE]
where is the maximum of the variances in the profile . Moreover, for any and all ,
[TABLE]
where is the Stieltjes transform of .
To prove this result, we use the following result of Gaussian concentration inequality for Lipschitz functions (see e.g. [18, Theorem 5.6]).
Theorem 7.4**.**
Let be a vector of independent standard normal random variables. Let be a -Lipschitz function for the Euclidean norm of . Then for all we have
[TABLE]
Proof.
We let be the subset of Hermitian matrices. For two unit vectors and any in we consider the function
[TABLE]
In order to use Theorem 7.4 for the real and the imaginary parts of , we shall estimate its Lipschitz constant. We denote by \big{\|}A\big{\|}_{F}=\big{(}\mathrm{Tr}\big{[}A^{*}A\big{]}\big{)}^{\frac{1}{2}} the Frobenius norm of a matrix . We use the isomorphism between endowed with and endowed as the Euclidean norm. Using Cauchy-Schwarz’s inequality and Lemma 5.3, it follows that, for any in ,
[TABLE]
Since can be decomposed in the basis (5.9) of Hermitian matrices as where the are reals, we have that is a -Lipschitz function (with Lipschitz constant ) of the matrix having real entries. Now, recall the decomposition (5.10) of into the basis (5.9)
[TABLE]
where the are i.i.d. real Gaussian random variables that are centered and of variance one. By Inequality (7.6) and after a change of variable, we get that the function is a -Lipschitz function (that is complex-valued) with Lipschitz constant
[TABLE]
Hence (7.3) follows by Gaussian concentration inequality, namely Equation (7.5). Now, let and recall that the Stieltjes transform is the map
[TABLE]
and let us consider the mapping A\mapsto\phi_{\lambda}(A):=\mathrm{Tr}\big{[}(\lambda\mathbb{I}_{N}-A-Y_{N})^{-1}\big{]} for . For any in , we remark that
[TABLE]
To obtain an appropriate upper bound for (7.8), we use the following inequalities (see e.g. Lemma II.2 in [42]): denoting by the singular values of ,
[TABLE]
which hold for any and . For two such matrices, Inequality (7.9) combined with Cauchy-Schwarz’s inequality implies that
[TABLE]
Since and (by the same argument as for the imaginary part in the proof of Lemma 5.1), one finally obtains that
[TABLE]
Using the fact , one obtains by similar arguments that
[TABLE]
which finally yields
[TABLE]
Hence, combining the above inequality with (7.8) and Lemma 5.3, it follows that
[TABLE]
Therefore, thanks to the decomposition (7.7) for , the mapping \tilde{X}_{N}\mapsto\frac{1}{N}\mathrm{Tr}\big{[}(\lambda\mathbb{I}_{N}-X_{N}-Y_{N})^{-1}\big{]}=\frac{1}{N}\phi_{\lambda}(X_{N}) is a -Lipschitz function with Lipschitz constant
[TABLE]
Therefore, using again Gaussian concentration for Lipschitz functions, one obtains Inequality (7.4), which completes the proof of Lemma 7.3. ∎
Now, let us fix , and consider satisfying so that \|\mathbb{E}\big{[}G_{H_{N}}(\Lambda)\big{]}-G^{\square}_{H_{N}}(\Lambda)\|\leqslant t_{N}^{(1)}, by Lemma 4.1, where
[TABLE]
Since and its expectation are diagonal matrices one has that the operator norm of their difference satisfies
[TABLE]
Thus, combining Inequality (7.3) with a union bound yields the following concentration inequality: for all ,
[TABLE]
Hence, taking (for some ), one finally obtains that
[TABLE]
which proves Inequality (1.14), and completes the proof of Theorem 1.1 in the general case. The case diagonal where is diagonal follows similarly.
Then, to derive the proof of Corollary 1.2, we use the concentration inequality (7.4) for the Stieltjes transform . Since g_{H_{N}}^{\square}(\lambda)=\frac{1}{N}\mathrm{Tr}\big{[}G_{H_{N}}^{\square}(\lambda\mathbb{I}_{N})\big{]} and given that |\frac{1}{N}\mathrm{Tr}\big{[}A\big{]}|\leqslant\|A\| for any diagonal matrix , we obtain from Lemma 4.1 that, for satisfying Condition (1.16),
[TABLE]
Therefore, taking (for some ) one obtains, by combining Inequalities (7.4) and (7.11), that
[TABLE]
which proves Inequality (1.17). The case diagonal where is diagonal also follows similarly, and this completes the proof of Corollary 1.2.
We now prove Corollary 1.3, assuming hence that is a diagonal matrix. We take with satisfying . Since is supposed to be Hermitian, it is a diagonal matrix with real entries. Therefore, by Corollary 7.2, one has that \mathbb{E}\big{[}(\lambda\mathbb{I}_{N}-H_{N})^{-1}\big{]} is diagonal. Then, we remark that
[TABLE]
Now, with \hat{t}_{N}^{(1)}:=\Big{(}1+\frac{\gamma^{2}_{\mathrm{max}}}{\delta|\Im m\,\lambda|^{2}}\Big{)}\frac{\gamma_{\mathrm{max}}^{4}}{N^{3/2}(\Im m\,\lambda)^{5}}, we have
[TABLE]
Moreover, if we denote by the columns of the matrix , we have
[TABLE]
Thus, by combining Inequality (7.3) with the fact that \mathbb{E}\big{[}(\lambda\mathbb{I}_{N}-H_{N})^{-1}\big{]}=\mathbb{E}\big{[}G_{H_{N}}(\lambda\mathbb{I}_{N})\big{]} (since \mathbb{E}\big{[}(\lambda\mathbb{I}_{N}-H_{N})^{-1}\big{]} is diagonal), it follows by a union bound argument that, for all ,
[TABLE]
Therefore, combining Inequalities (7.12), (7.13) and (7.14) we obtain that
[TABLE]
with , which finally yields Inequality (1.19).
Let us now prove Corollary 1.4, where is not necessarily diagonal. In particular \mathbb{E}\big{[}(\lambda\mathbb{I}_{N}-H_{N})^{-1}\big{]} is not necessarily a diagonal matrix. Thus, we shall use the deterministic equivalent defined by (1.21) to approximate . First, as previously, we remark that
[TABLE]
where \Omega_{H_{N}}^{\square}(\lambda\mathbb{I}_{N})=\lambda\mathbb{I}_{N}-\mathcal{R}_{N}\big{(}G_{H_{N}}^{\square}(\lambda\mathbb{I}_{N})\big{)}. By the same arguments used to derive inequality (7.14), it follows that
[TABLE]
Hence, the only difference with the setting where is diagonal is the control of the term \|U_{N,k}^{*}\left(\mathbb{E}\big{[}(\lambda\mathbb{I}_{N}-H_{N})^{-1}\big{]}-\big{(}\Omega_{H_{N}}^{\square}(\lambda\mathbb{I}_{N})-Y_{N}\big{)}^{-1}\right)U_{N,k}\| which is obviously bounded by \|\mathbb{E}\big{[}(\lambda\mathbb{I}_{N}-H_{N})^{-1}\big{]}-\big{(}\Omega_{H_{N}}^{\square}(\lambda\mathbb{I}_{N})-Y_{N}\big{)}^{-1}\|. We now consider the decomposition
[TABLE]
where \big{(}\Omega_{H_{N}}(\lambda\mathbb{I}_{N})-Y_{N}\big{)}^{-1}=\lambda\mathbb{I}_{N}-\mathcal{R}_{N}\Big{(}\mathbb{E}\big{[}G_{H_{N}}(\lambda\mathbb{I}_{N})\big{]}\Big{)}. By combining Lemma 5.3, Lemma 5.7 and Lemma 5.8, we obtain that
[TABLE]
Now, using the equality \big{(}\Omega_{H_{N}}(\lambda\mathbb{I}_{N})-Y_{N}\big{)}^{-1}-\big{(}\Omega_{H_{N}}^{\square}(\lambda\mathbb{I}_{N})-Y_{N}\big{)}^{-1}=\big{(}\Omega_{H_{N}}^{\square}(\lambda\mathbb{I}_{N})-Y_{N}\big{)}^{-1}\left(\Omega_{H_{N}}^{\square}(\lambda\mathbb{I}_{N})-\Omega_{H_{N}}(\lambda\mathbb{I}_{N})\right)\big{(}\Omega_{H_{N}}(\lambda\mathbb{I}_{N})-Y_{N}\big{)}^{-1}, one has that
[TABLE]
By Lemma 5.3 and Lemma 5.7 one obtains that \|\big{(}\Omega_{H_{N}}(\lambda\mathbb{I}_{N})-Y_{N}\big{)}^{-1}\|\leqslant(\Im m\,\lambda)^{-1}. Using again Lemma 5.3 one has that \|\big{(}\Omega_{H_{N}}^{\square}(\lambda\mathbb{I}_{N})-Y_{N}\big{)}^{-1}~\|\leqslant\|\big{(}\Im m\,\Omega_{H_{N}}^{\square}(\lambda\mathbb{I}_{N})\big{)}^{-1}\|. Since \Omega_{H_{N}}^{\square}(\lambda\mathbb{I}_{N})=\lambda\mathbb{I}_{N}-\mathcal{R}_{N}\big{(}G_{H_{N}}^{\square}(\lambda\mathbb{I}_{N})\big{)} and given that by Lemma 6.3, it follows from Lemma 5.4 that \Im m\big{(}\mathcal{R}_{N}\big{(}G_{H_{N}}^{\square}(\lambda\mathbb{I}_{N})\big{)}\big{)}\leqslant 0. Consequently, one has that , and this finally yields \|\big{(}\Omega_{H_{N}}^{\square}(\lambda\mathbb{I}_{N})-Y_{N}\big{)}^{-1}~\|\leqslant(\Im m\,\lambda)^{-1}.
Then, using Lemma 5.4, we remark that
[TABLE]
and therefore, if , then Lemma 4.1 implies that
[TABLE]
Therefore, we obtain that
[TABLE]
Hence, the use of Inequalities (7.17) and (7.18) yields that
[TABLE]
Finally, combining Inequalities (7.15), (7.16) and (7.19) yields Inequality (1.22), which completes the proofs of the results stated in Section 1.2.
7.3 Convergence of the fixed point algorithm
A key step in these numerical experiments in the numerical approximation of the solution of the fixed point equation (1.12) through an iterative algorithm whose convergence is first briefly discussed.
Lemma 7.5**.**
For any diagonal function , we consider the sequence of diagonal functions \big{(}G^{(n)}_{N}\big{)}_{n\geqslant 0} given by G^{(n+1)}_{N}(\Lambda)=\psi_{\Lambda}\big{(}G^{(n)}_{N}(\Lambda)\big{)} for any . Then, as goes to infinity, the sequence converges to , where denotes the deterministic equivalent characterized in Lemma 6.3.
Proof.
Recall that by Lemma 6.1 the sequence \big{(}G^{(n)}_{N}(\Lambda)\big{)}_{n\geqslant 0} is bounded, and so up to a subsequence it converges to some diagonal matrix . If , by Corollary 6.2 then by contractivity of the fixed point problem. Moreover, for each , the function is an analytic function of the variable , uniformly bounded for for any . By dominated convergence, the limit up to any subsequence is analytic. By analytic continuation, all limit coincide and are equal to . ∎
The reference list from the paper itself. Each links out to its DOI / PubMed record.
- 1[1] O. Ajanki, T. Krüger, and L. Erdős , Singularities of solutions to quadratic vector equations on the complex upper half-plane , Communications on Pure and Applied Mathematics, 70 (2017), pp. 1672–1705.
- 2[2] O. H. Ajanki, L. Erdős, and T. Krüger , Universality for general Wigner-type matrices , Probability Theory and Related Fields, 169 (2017), pp. 667–727.
- 3[3] , Stability of the matrix Dyson equation and random matrices with correlations , Probability Theory and Related Fields, 173 (2019), pp. 293–373.
- 4[4] J. Alt, L. Erdős, and T. Krüger , Local law for random Gram matrices , Electron. J. Probab., 22 (2017), p. 41 pp.
- 5[5] , Local inhomogeneous circular law , Ann. Appl. Probab., 28 (2018), pp. 148–203.
- 6[6] J. Alt, L. Erdös, T. Krüger, and Y. Nemish , Location of the spectrum of Kronecker random matrices , Ann. Inst. H. Poincaré Probab. Statist., 55 (2019), pp. 661–696.
- 7[7] B. Au, G. Cébron, A. Dahlqvist, F. Gabriel, and C. Male , Large permutation invariant random matrices are asymptotically free over the diagonal , ar Xiv e-prints, (2018), p. ar Xiv:1805.07045.
- 8[8] Z. Bai and J. Silverstein , Spectral Analysis of Large Dimensional Random Matrices , Springer Series in Statistics, Springer New York, 2012.