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68
Covariance regularization by thresholding
, 2007
"... This paper considers regularizing a covariance matrix of p variables estimated from n observations, by hard thresholding. We show that the thresholded estimate is consistent in the operator norm as long as the true covariance matrix is sparse in a suitable sense, the variables are Gaussian or subGa ..."
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Cited by 148 (11 self)
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This paper considers regularizing a covariance matrix of p variables estimated from n observations, by hard thresholding. We show that the thresholded estimate is consistent in the operator norm as long as the true covariance matrix is sparse in a suitable sense, the variables are Gaussian or subGaussian, and (log p)/n → 0, and obtain explicit rates. The results are uniform over families of covariance matrices which satisfy a fairly natural notion of sparsity. We discuss an intuitive resampling scheme for threshold selection and prove a general crossvalidation result that justifies this approach. We also compare thresholding to other covariance estimators in simulations and on an example from climate data. 1. Introduction. Estimation
Operator norm consistent estimation of largedimensional sparse covariance matrices
 Annals of Statistics
"... Estimating covariance matrices is a problem of fundamental importance in multivariate statistics. In practice it is increasingly frequent to work with data matrices X of dimension n×p, where p and n are both large. Results from random matrix theory show very clearly that in this setting, standard es ..."
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Cited by 69 (1 self)
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Estimating covariance matrices is a problem of fundamental importance in multivariate statistics. In practice it is increasingly frequent to work with data matrices X of dimension n×p, where p and n are both large. Results from random matrix theory show very clearly that in this setting, standard estimators like the sample covariance matrix perform in general very poorly. In this “large n, large p ” setting, it is sometimes the case that practitioners are willing to assume that many elements of the population covariance matrix are equal to 0, and hence this matrix is sparse. We develop an estimator to handle this situation. The estimator is shown to be consistent in operator norm, when, for instance, we have p ≍ n as n → ∞. In other words the largest singular value of the difference between the estimator and the population covariance matrix goes to zero. This implies consistency of all the eigenvalues and consistency of eigenspaces associated to isolated eigenvalues. We also propose a notion of sparsity for matrices, that is, “compatible” with spectral analysis and is independent of the ordering of the variables. 1. Introduction. Estimating
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 67 (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.
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 66 (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 66 (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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High dimensional statistical inference and random matrices
 IN: PROCEEDINGS OF INTERNATIONAL CONGRESS OF MATHEMATICIANS
, 2006
"... Multivariate statistical analysis is concerned with observations on several variables which are thought to possess some degree of interdependence. Driven by problems in genetics and the social sciences, it first flowered in the earlier half of the last century. Subsequently, random matrix theory ..."
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Cited by 49 (1 self)
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Multivariate statistical analysis is concerned with observations on several variables which are thought to possess some degree of interdependence. Driven by problems in genetics and the social sciences, it first flowered in the earlier half of the last century. Subsequently, random matrix theory (RMT) developed, initially within physics, and more recently widely in mathematics. While some of the central objects of study in RMT are identical to those of multivariate statistics, statistical theory was slow to exploit the connection. However, with vast data collection ever more common, data sets now often have as many or more variables than the number of individuals observed. In such contexts, the techniques and results of RMT have much to offer multivariate statistics. The paper reviews some of the progress to date.
Sample eigenvalue based detection of highdimensional signals in white noise using relatively few samples
, 2007
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