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187
Eigenvalues of large sample covariance matrices of spiked population models
, 2006
"... We consider a spiked population model, proposed by Johnstone, whose population eigenvalues are all unit except for a few fixed eigenvalues. The question is to determine how the sample eigenvalues depend on the nonunit population ones when both sample size and population size become large. This pape ..."
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Cited by 163 (8 self)
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We consider a spiked population model, proposed by Johnstone, whose population eigenvalues are all unit except for a few fixed eigenvalues. The question is to determine how the sample eigenvalues depend on the nonunit population ones when both sample size and population size become large. This paper completely determines the almost sure limits for a general class of samples. 1
Scaling limit for the spacetime covariance of the stationary totally asymmetric simple exclusion process
 Comm. Math. Phys
"... The totally asymmetric simple exclusion process (TASEP) on the onedimensional lattice with the Bernoulli ρ measure as initial conditions, 0 < ρ < 1, is stationary in space and time. Let Nt(j) be the number of particles which have crossed the bond from j to j + 1 during the time span [0,t]. Fo ..."
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Cited by 83 (30 self)
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The totally asymmetric simple exclusion process (TASEP) on the onedimensional lattice with the Bernoulli ρ measure as initial conditions, 0 < ρ < 1, is stationary in space and time. Let Nt(j) be the number of particles which have crossed the bond from j to j + 1 during the time span [0,t]. For j = (1 − 2ρ)t + 2w(ρ(1 − ρ)) 1/3 t 2/3 we prove that the fluctuations of Nt(j) for large t are of order t 1/3 and we determine the limiting distribution function Fw(s), which is a generalization of the GUE TracyWidom distribution. The family Fw(s) of distribution functions have been obtained before by Baik and Rains in the context of the PNG model with boundary sources, which requires the asymptotics of a RiemannHilbert problem. In our work we arrive at Fw(s) through the asymptotics of a Fredholm determinant. Fw(s) is simply related to the scaling function for the spacetime covariance of the stationary TASEP, equivalently to the asymptotic transition
Beta ensembles, stochastic Airy spectrum, and a diffusion
, 2008
"... We prove that the largest eigenvalues of the beta ensembles of random matrix theory converge in distribution to the lowlying eigenvalues of the random Schrödinger operator − d2 dx 2 + x + 2 √ β b ′ x restricted to the positive halfline, where b ′ x is white noise. In doing so we extend the definit ..."
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Cited by 69 (9 self)
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We prove that the largest eigenvalues of the beta ensembles of random matrix theory converge in distribution to the lowlying eigenvalues of the random Schrödinger operator − d2 dx 2 + x + 2 √ β b ′ x restricted to the positive halfline, where b ′ x is white noise. In doing so we extend the definition of the TracyWidom(β) distributions to all β> 0, and also analyze their tails. Last, in a parallel development, we provide a second characterization of these laws in terms of a onedimensional diffusion. The proofs rely on the associated tridiagonal matrix models and a universality result showing that the spectrum of such models converge to that of their continuum operator limit. In particular, we show how TracyWidom laws arise from a functional central limit theorem.
TracyWidom limit for the largest eigenvalue of a large class of complex sample covariance matrices
 ANN. PROBAB
, 2007
"... We consider the asymptotic fluctuation behavior of the largest eigenvalue of certain sample covariance matrices in the asymptotic regime where both dimensions of the corresponding data matrix go to infinity. More precisely, let X be an n × p matrix, and let its rows be i.i.d. complex normal vectors ..."
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Cited by 68 (6 self)
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We consider the asymptotic fluctuation behavior of the largest eigenvalue of certain sample covariance matrices in the asymptotic regime where both dimensions of the corresponding data matrix go to infinity. More precisely, let X be an n × p matrix, and let its rows be i.i.d. complex normal vectors with mean 0 and covariance �p. We show that for a large class of covariance matrices �p, the largest eigenvalue of X ∗ X is asymptotically distributed (after recentering and rescaling) as the Tracy–Widom distribution that appears in the study of the Gaussian unitary ensemble. We give explicit formulas for the centering and scaling sequences that are easy to implement and involve only the spectral distribution of the population covariance, n and p. The main theorem applies to a number of covariance models found in applications. For example, wellbehaved Toeplitz matrices as well as covariance matrices whose spectral distribution is a sum of atoms (under some conditions on the mass of the atoms) are among the models the theorem can handle. Generalizations of the theorem to certain spiked versions of our models and a.s. results about the largest eigenvalue are given. We also discuss a simple corollary that does not require normality of the entries of the data matrix and some consequences for applications in multivariate statistics.
SPECTRUM ESTIMATION FOR LARGE DIMENSIONAL COVARIANCE MATRICES USING RANDOM MATRIX THEORY
 SUBMITTED TO THE ANNALS OF STATISTICS
"... Estimating the eigenvalues of a population covariance matrix from a sample covariance matrix is a problem of fundamental importance in multivariate statistics; the eigenvalues of covariance matrices play a key role in many widely techniques, in particular in Principal Component Analysis (PCA). In ma ..."
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Cited by 65 (4 self)
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Estimating the eigenvalues of a population covariance matrix from a sample covariance matrix is a problem of fundamental importance in multivariate statistics; the eigenvalues of covariance matrices play a key role in many widely techniques, in particular in Principal Component Analysis (PCA). In many modern data analysis problems, statisticians are faced with large datasets where the sample size, n, is of the same order of magnitude as the number of variables p. Random matrix theory predicts that in this context, the eigenvalues of the sample covariance matrix are not good estimators of the eigenvalues of the population covariance. We propose to use a fundamental result in random matrix theory, the MarčenkoPastur equation, to better estimate the eigenvalues of large dimensional covariance matrices. The MarčenkoPastur equation holds in very wide generality and under weak assumptions. The estimator we obtain can be thought of as “shrinking ” in a non linear fashion the eigenvalues of the sample covariance matrix to estimate the population eigenvalues. Inspired by ideas of random matrix theory, we also suggest a change of point of view when thinking about estimation of highdimensional vectors: we do not try to estimate directly the vectors but rather a probability measure that describes them. We think this is a theoretically more fruitful way to think about these problems. Our estimator gives fast and good or very good results in extended simulations. Our algorithmic approach is based on convex optimization. We also show that the proposed estimator is consistent.
FINITE SAMPLE APPROXIMATION RESULTS FOR PRINCIPAL COMPONENT ANALYSIS: A MATRIX PERTURBATION APPROACH
"... Principal Component Analysis (PCA) is a standard tool for dimensional reduction of a set of n observations (samples), each with p variables. In this paper, using a matrix perturbation approach, we study the nonasymptotic relation between the eigenvalues and eigenvectors of PCA computed on a finite ..."
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Cited by 65 (15 self)
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Principal Component Analysis (PCA) is a standard tool for dimensional reduction of a set of n observations (samples), each with p variables. In this paper, using a matrix perturbation approach, we study the nonasymptotic relation between the eigenvalues and eigenvectors of PCA computed on a finite sample of size n, to those of the limiting population PCA as n → ∞. As in machine learning, we present a finite sample theorem which holds with high probability for the closeness between the leading eigenvalue and eigenvector of sample PCA and population PCA under a spiked covariance model. In addition, we also consider the relation between finite sample PCA and the asymptotic results in the joint limit p, n → ∞, with p/n = c. We present a matrix perturbation view of the “phase transition phenomenon”, and a simple linearalgebra based derivation of the eigenvalue and eigenvector overlap in this asymptotic limit. Moreover, our analysis also applies for finite p, n where we show that although there is no sharp phase transition as in the infinite case, either as a function of noise level or as a function of sample size n, the eigenvector of sample PCA may exhibit a sharp ”loss of tracking”, suddenly losing its relation to the (true) eigenvector of the population PCA matrix. This occurs due to a crossover between the eigenvalue due to the signal and the largest eigenvalue due to noise, whose eigenvector points in a random direction.
The eigenvalues and eigenvectors of finite, low rank perturbations of large random matrices
, 2011
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The largest eigenvalue of small rank perturbations of Hermitian random matrices, Prob
 Theor. Rel. Fields
"... 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 Mij, i ≤ j with various expectation ..."
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Cited by 58 (1 self)
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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 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 (EMij) N i,j=1 and its rank. 1 Introduction and
The largest eigenvalue of finite rank deformation of large Wigner matrices: convergence and nonuniversality of the fluctuations
, 2007
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The largest eigenvalue of rank one deformation of large Wigner matrices
 COMM. MATH. PHYS
, 2008
"... 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 (c.f. [12]) in the investigations of class ..."
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Cited by 50 (4 self)
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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 (c.f. [12]) 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.