arXiv subjects

Sandrine Péché

Publications and source records attributed to Sandrine Péché.

16 recordsLinked to original sources

arXiv2022

Largest Eigenvalues of the Conjugate Kernel of Single-Layered Neural Networks

This paper is concerned with the asymptotic distribution of the largest eigenvalues for some nonlinear random matrix ensemble stemming from the study of neural networks. More precisely we consider $M= \frac{1}{m} YY^\top$ with $Y=f(WX)$ where $W$ and $X$ are random rectangular matrices with i.i.d. centered entries. This models the data covariance matrix or the Conjugate Kernel of a single layered random Feed-Forward Neural Network. The function $f$ is applied entrywise and can be seen as the activation function of the neural network. We show that the largest eigenvalue has the same limit (in probability) as that of some well-known linear random matrix ensembles. In particular, we relate the asymptotic limit of the largest eigenvalue for the nonlinear model to that of an information-plus-noise random matrix, establishing a possible phase transition depending on the function $f$ and the distribution of $W$ and $X$. This may be of interest for applications to machine learning.

math.PR↗

arXiv2020

Joint Global Fluctuations of complex Wigner and deterministic Matrices

We characterize the limiting fluctuations of traces of several independent Wigner matrices and deterministic matrices under mild conditions. A CLT holds but in general the families are not asymptotically free of second order and the limiting covariance depends on more information on the deterministic matrices than their limiting *-distribution.

math.PR↗

arXiv2020

Eigenvalue distribution of nonlinear models of random matrices

This paper is concerned with the asymptotic empirical eigenvalue distribution of a non linear random matrix ensemble. More precisely we consider $M= \frac{1}{m} YY^*$ with $Y=f(WX)$ where $W$ and $X$ are random rectangular matrices with i.i.d. centered entries. The function $f$ is applied pointwise and can be seen as an activation function in (random) neural networks. We compute the asymptotic empirical distribution of this ensemble in the case where $W$ and $X$ have sub-Gaussian tails and $f$ is real analytic. This extends a previous result where the case of Gaussian matrices $W$ and $X$ is considered. We also investigate the same questions in the multi-layer case, regarding neural network applications.

math.PR↗

arXiv2015

Localization and delocalization for heavy tailed band matrices

We consider some random band matrices with band-width $N^μ$ whose entries are independent random variables with distribution tail in $x^{-α}$. We consider the largest eigenvalues and the associated eigenvectors and prove the following phase transition. On the one hand, when $α\textless{}2(1+μ^{-1})$, the largest eigenvalues have order $N^{(1+μ)/α}$, are asymptotically distributed as a Poisson process and their associated eigenvectors are essentially carried by two coordinates (this phenomenon has already been remarked by Soshnikov for full matrices with heavy tailed entries,i.e. when $α\textless{}2$, and by Auffinger, Ben Arous and P{é}ch{é} when $α\textless{}4$). On the other hand, when $α\textgreater{}2(1+μ^{-1})$, the largest eigenvalues have order $N^{μ/2}$ and most eigenvectors of the matrix are delocalized, i.e. approximately uniformly distributed on their $N$ coordinates.

math.PR↗

arXiv2015

Poisson statistics for matrix ensembles at large temperature

In this article, we consider $β$-ensembles, i.e. collections of particles with random positions on the real line having joint distribution $$\frac{1}{Z_N(β)}|Δ(λ)|^βe^{- \frac{Nβ}{4}\sum_{i=1}^Nλ_i^2}d λ,$$ in the regime where $β\to 0$ as $N\to\infty$. We briefly describe the global regime and then consider the local regime. In the case where $Nβ$ stays bounded, we prove that the local eigenvalue statistics, in the vicinity of any real number, are asymptotically to those of a Poisson point process. In the case where $Nβ\to\infty$, we prove a partial result in this direction.

math.PR↗

arXiv2014

Uniform regular weighted graphs with large degree: Wigner's law, asymptotic freeness and graphons limit

For each $N\geq 1$, let $G_N$ be a simple random graph on the set of vertices $[N]=\{1,2, ..., N\}$, which is invariant by relabeling of the vertices. The asymptotic behavior as $N$ goes to infinity of correlation functions: $$ \mathfrak C_N(T)= \mathbb E\bigg[ \prod_{(i,j) \in T} \Big(\mathbf 1_{\big(\{i,j\} \in G_N \big)} - \mathbb P(\{i,j\} \in G_N) \Big)\bigg], \ T \subset [N]^2 \textrm{finite}$$ furnishes informations on the asymptotic spectral properties of the adjacency matrix $A_N$ of $G_N$. Denote by $d_N = N\times \mathbb P(\{i,j\} \in G_N) $ and assume $d_N, N-d_N\underset{N \rightarrow \infty}{\longrightarrow} \infty$. If $\mathfrak C_N(T) =\big(\frac{d_N}N\big)^{|T|} \times O\big(d_N^{-\frac {|T|}2}\big)$ for any $T$, the standardized empirical eigenvalue distribution of $A_N$ converges in expectation to the semicircular law and the matrix satisfies asymptotic freeness properties in the sense of free probability theory. We provide such estimates for uniform $d_N$-regular graphs $G_{N,d_N}$, under the additional assumption that $|\frac N 2 - d_N- η\sqrt{d_N}| \underset{N \rightarrow \infty}{\longrightarrow} \infty$ for some $η>0$. Our method applies also for simple graphs whose edges are labelled by i.i.d. random variables.

math.PR↗

arXiv2014

A note on the second order universality at the edge of Coulomb gases on the plane

We consider in this note a class of two-dimensional determinantal Coulomb gases confined by a radial external field. As the number of particles tends to infinity, their empirical distribution tends to a probability measure supported in a centered ring of the complex plane. A quadratic confinement corresponds to the complex Ginibre Ensemble. In this case, it is also already known that the asymptotic fluctuation of the radial edge follows a Gumbel law. We establish in this note the universality of this edge behavior, beyond the quadratic case. The approach, inspired by earlier works of Kostlan and Rider, boils down to identities in law and to an instance of the Laplace method.

math.PR↗

arXiv2014

Largest eigenvalues and eigenvectors of band or sparse matrices

In this text, we consider an N by N random matrix X such that all but o(N) rows of X have W non identically zero entries, the other rows having lass than $W$ entries (such as, for example, standard or cyclic band matrices). We always suppose that 1 << W << N. We first prove that if the entries are independent, centered, have variance one, satisfy a certain tail upper-bound condition and W >> (\log N)^{6(1+α)}, where αis a positive parameter depending on the distribution of the entries, then the largest eigenvalue of X/\sqrt{W} converges to the upper bound of its limit spectral distribution, that is 2, as for Wigner matrices. This extends some previous results by Khorunzhiy and Sodin where less hypotheses were made on W, but more hypotheses were made about the law of the entries and the structure of the matrix. Then, under the same hypotheses, we prove a delocalization result for the eigenvectors of X, precisely that most of them cannot be essentially localized on less than W/\log(N) entries. This lower bound on the localization length has to be compared to the recent result by Steinerberger, which states that either the localization length in the edge is << W^{7/5} or there is strong interaction between two eigenvectors in an interval of length W^{7/5}.

math.PR↗

arXiv2013

Convergence of the two-point function of the stationary TASEP

We consider the two-point function of the totally asymmetric simple exclusion process with stationary initial conditions. The two-point function can be expressed as the discrete Laplacian of the variance of the associated height function. The limit of the distribution function of the appropriately scaled height function was obtained previously by Ferrari and Spohn. In this paper we show that the convergence can be improved to the convergence of moments. This implies the convergence of the two-point function in a weak sense along the near-characteristic direction as time tends to infinity, thereby confirming the conjecture in the paper of Ferrari and Spohn.

math-ph↗

arXiv2012

Limit process of stationary TASEP near the characteristic line

The totally asymmetric simple exclusion process (TASEP) on Z with the Bernoulli-rho measure as initial conditions, 0<rho<1, is stationary. It is known that along the characteristic line, the current fluctuates as of order t^{1/3}. The limiting distribution has also been obtained explicitly. In this paper we determine the limiting multi-point distribution of the current fluctuations moving away from the characteristics by the order t^{2/3}. The main tool is the analysis of a related directed last percolation model. We also discuss the process limit in tandem queues in equilibrium.

math-ph↗

arXiv2010

Limit processes for TASEP with shocks and rarefaction fans

We consider the totally asymmetric simple exclusion process (TASEP) with two-sided Bernoulli initial condition, i.e., with left density rho_- and right density rho_+. We consider the associated height function, whose discrete gradient is given by the particle occurrences. Macroscopically one has a deterministic limit shape with a shock or a rarefaction fan depending on the values of rho_{+/-}. We characterize the large time scaling limit of the fluctuations as a function of the densities rho_{+/-} and of the different macroscopic regions. Moreover, using a slow decorrelation phenomena, the results are extended from fixed time to the whole space-time, except along the some directions (the characteristic solutions of the related Burgers equation) where the problem is still open. On the way to proving the results for TASEP, we obtain the limit processes for the fluctuations in a class of corner growth processes with external sources, of equivalently for the last passage time in a directed percolation model with two-sided boundary conditions. Additionally, we provide analogous results for eigenvalues of perturbed complex Wishart (sample covariance) matrices.

math-ph↗

arXiv2009

Eigenvectors of some large sample covariance matrix ensembles

We consider sample covariance matrices $S_N=\frac{1}{p}Σ_N^{1/2}X_NX_N^* Σ_N^{1/2}$ where $X_N$ is a $N \times p$ real or complex matrix with i.i.d. entries with finite $12^{\rm th}$ moment and $Σ_N$ is a $N \times N$ positive definite matrix. In addition we assume that the spectral measure of $Σ_N$ almost surely converges to some limiting probability distribution as $N \to \infty$ and $p/N \to γ>0.$ We quantify the relationship between sample and population eigenvectors by studying the asymptotics of functionals of the type $\frac{1}{N} \text{Tr} (g(Σ_N) (S_N-zI)^{-1})),$ where $I$ is the identity matrix, $g$ is a bounded function and $z$ is a complex number. This is then used to compute the asymptotically optimal bias correction for sample eigenvalues, paving the way for a new generation of improved estimators of the covariance matrix and its inverse.

math.PR↗

arXiv2008

The largest eigenvalues of sample covariance matrices for a spiked population: diagonal case

We consider large complex random sample covariance matrices obtained from "spiked populations", that is when the true covariance matrix is diagonal with all but finitely many eigenvalues equal to one. We investigate the limiting behavior of the largest eigenvalues when the population and the sample sizes both become large. Under some conditions on moments of the sample distribution, we prove that the asymptotic fluctuations of the largest eigenvalues are the same as for a complex Gaussian sample with the same true covariance. The real setting is also considered.

math-ph↗

arXiv2007

Large deviations for weighted empirical mean with outliers

We study in this article large deviations for the empirical mean of iid random vectors with some deterministic weights, whose empirical measure weakly converges to some compactly support probability distribution. The scope of this paper is to study the effect on the LDP of outliers, that is sequences of weights that remain far from the support of the limiting measure.

math.PR↗

arXiv2006

The largest eigenvalue of rank one deformation of large Wigner matrices

The purpose of this paper is to establish universality of the fluctuations of the largest eigenvalue of some non necessarily Gaussian complex Deformed Wigner Ensembles. The real model is also considered. Our approach is close to the one used by A. Soshnikov in the investigations of classical real or complex Wigner Ensembles. It is based on the computation of moments of traces of high powers of the random matrices under consideration.

math.PR↗

arXiv2005

The largest eigenvalue of small rank perturbations of Hermitian random matrices

We compute the limiting eigenvalue statistics at the edge of the spectrum of large Hermitian random matrices perturbed by the addition of small rank deterministic matrices. To be more precise, we consider random Hermitian matrices with independent Gaussian entries $M_{ij}, i\leq j$ with various expectations. We prove that the largest eigenvalue of such random matrices exhibits, in the large $N$ limit, various limiting distributions depending on both the eigenvalues of the matrix $(\mathbb{E}M_{ij})_{i,j=1}^N$ and its rank.

math.PR↗