Sandrine Péché

Bio: Sandrine Péché is an academic researcher from University of Paris. The author has contributed to research in topics: Random matrix & Eigenvalues and eigenvectors. The author has an hindex of 22, co-authored 41 publications receiving 2512 citations. Previous affiliations of Sandrine Péché include University of California, Davis & Joseph Fourier University.

Phase transition of the largest eigenvalue for non-null complex sample covariance matrices

Abstract: We compute the limiting distributions of the largest eigenvalue of a complex Gaussian sample covariance matrix when both the number of samples and the number of variables in each sample become large. When all but finitely many, say $r$, eigenvalues of the covariance matrix are the same, the dependence of the limiting distribution of the largest eigenvalue of the sample covariance matrix on those distinguished $r$ eigenvalues of the covariance matrix is completely characterized in terms of an infinite sequence of new distribution functions that generalize the Tracy-Widom distributions of the random matrix theory. Especially a phase transition phenomena is observed. Our results also apply to a last passage percolation model and a queuing model.

The Largest Eigenvalue of Rank One Deformation of Large Wigner Matrices

Abstract: The purpose of this paper is to establish universality of the fluctuations of the largest eigenvalue for 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 (cf. [11]) 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.

The largest eigenvalue of small rank perturbations of Hermitian random matrices

Abstract: 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. We consider random Hermitian matrices with independent Gaussian entries Mij,i≤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 Open image in new window and its rank. This rank is also allowed to increase with N in some restricted way.

Eigenvectors of some large sample covariance matrix ensembles

Abstract: We consider sample covariance matrices $${S_N=\frac{1}{p}\Sigma_N^{1/2}X_NX_N^* \Sigma_N^{1/2}}$$ where X N is a N × p real or complex matrix with i.i.d. entries with finite 12th moment and Σ N is a N × 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 → ∞ and p/N → γ > 0. We quantify the relationship between sample and population eigenvectors by studying the asymptotics of functionals of the type $${\frac{1}{N}\text{Tr} ( g(\Sigma_N) (S_N-zI)^{-1}),}$$ 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.

Universality of local eigenvalue statistics for some sample covariance matrices

Abstract: We consider random, complex sample covariance matrices 1 X ∗ X, where X is a p× N random matrix with i.i.d. entries of distribution � . It has been conjectured that both the distribution of the distance between nearest neighbor eigenva lues in the bulk and that of the smallest eigenvalues become, in the limit N → ∞, p N → 1, the same as that identified for a complex Gaussian distribution � . We prove these conjectures for a certain class of probability distributions � . c 2004 Wiley Periodicals, Inc.

Principal components analysis corrects for stratification in genome-wide association studies

Abstract: Population stratification—allele frequency differences between cases and controls due to systematic ancestry differences—can cause spurious associations in disease studies. We describe a method that enables explicit detection and correction of population stratification on a genome-wide scale. Our method uses principal components analysis to explicitly model ancestry differences between cases and controls. The resulting correction is specific to a candidate marker’s variation in frequency across ancestral populations, minimizing spurious associations while maximizing power to detect true associations. Our simple, efficient approach can easily be applied to disease studies with hundreds of thousands of markers. Population stratification—allele frequency differences between cases and controls due to systematic ancestry differences—can cause spurious associations in disease studies 1‐8 . Because the effects of stratification vary in proportion to the number of samples 9 , stratification will be an increasing problem in the large-scale association studies of the future, which will analyze thousands of samples in an effort to detect common genetic variants of weak effect. The two prevailing methods for dealing with stratification are genomic control and structured association 9‐14 . Although genomic control and structured association have proven useful in a variety of contexts, they have limitations. Genomic control corrects for stratification by adjusting association statistics at each marker by a uniform overall inflation factor. However, some markers differ in their allele frequencies across ancestral populations more than others. Thus, the uniform adjustment applied by genomic control may be insufficient at markers having unusually strong differentiation across ancestral populations and may be superfluous at markers devoid of such differentiation, leading to a loss in power. Structured association uses a program such as STRUCTURE 15 to assign the samples to discrete subpopulation clusters and then aggregates evidence of association within each cluster. If fractional membership in more than one cluster is allowed, the method cannot currently be applied to genome-wide association studies because of its intensive computational cost on large data sets. Furthermore, assignments of individuals to clusters are highly sensitive to the number of clusters, which is not well defined 14,16 .

Population structure and eigenanalysis

Abstract: Current methods for inferring population structure from genetic data do not provide formal significance tests for population differentiation. We discuss an approach to studying population structure (principal components analysis) that was first applied to genetic data by Cavalli-Sforza and colleagues. We place the method on a solid statistical footing, using results from modern statistics to develop formal significance tests. We also uncover a general “phase change” phenomenon about the ability to detect structure in genetic data, which emerges from the statistical theory we use, and has an important implication for the ability to discover structure in genetic data: for a fixed but large dataset size, divergence between two populations (as measured, for example, by a statistic like FST) below a threshold is essentially undetectable, but a little above threshold, detection will be easy. This means that we can predict the dataset size needed to detect structure.

Introduction to the non-asymptotic analysis of random matrices.

Abstract: This is a tutorial on some basic non-asymptotic methods and concepts in random matrix theory. The reader will learn several tools for the analysis of the extreme singular values of random matrices with independent rows or columns. Many of these methods sprung off from the development of geometric functional analysis since the 1970's. They have applications in several fields, most notably in theoretical computer science, statistics and signal processing. A few basic applications are covered in this text, particularly for the problem of estimating covariance matrices in statistics and for validating probabilistic constructions of measurement matrices in compressed sensing. These notes are written particularly for graduate students and beginning researchers in different areas, including functional analysts, probabilists, theoretical statisticians, electrical engineers, and theoretical computer scientists.

An Introduction to Random Matrices

Abstract: The theory of random matrices plays an important role in many areas of pure mathematics and employs a variety of sophisticated mathematical tools (analytical, probabilistic and combinatorial). This diverse array of tools, while attesting to the vitality of the field, presents several formidable obstacles to the newcomer, and even the expert probabilist. This rigorous introduction to the basic theory is sufficiently self-contained to be accessible to graduate students in mathematics or related sciences, who have mastered probability theory at the graduate level, but have not necessarily been exposed to advanced notions of functional analysis, algebra or geometry. Useful background material is collected in the appendices and exercises are also included throughout to test the reader's understanding. Enumerative techniques, stochastic analysis, large deviations, concentration inequalities, disintegration and Lie algebras all are introduced in the text, which will enable readers to approach the research literature with confidence.

Topics in Random Matrix Theory

Abstract: The field of random matrix theory has seen an explosion of activity in recent years, with connections to many areas of mathematics and physics. However, this makes the current state of the field almost too large to survey in a single book. In this graduate text, we focus on one specific sector of the field, namely the spectral distribution of random Wigner matrix ensembles (such as the Gaussian Unitary Ensemble), as well as iid matrix ensembles. The text is largely self-contained and starts with a review of relevant aspects of probability theory and linear algebra. With over 200 exercises, the book is suitable as an introductory text for beginning graduate students seeking to enter the field.