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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 27 (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.
Concentration of measure and spectra of random matrices: with applications to correlation matrices, elliptical distributions and beyond
 THE ANNALS OF APPLIED PROBABILITY TO APPEAR
, 2009
"... We place ourselves in the setting of highdimensional statistical inference, where the number of variables p in a dataset of interest is of the same order of magnitude as the number of observations n. More formally we study the asymptotic properties of correlation and covariance matrices under the s ..."
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Cited by 8 (5 self)
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We place ourselves in the setting of highdimensional statistical inference, where the number of variables p in a dataset of interest is of the same order of magnitude as the number of observations n. More formally we study the asymptotic properties of correlation and covariance matrices under the setting that p/n → ρ ∈ (0, ∞), for general population covariance. We show that spectral properties for large dimensional correlation matrices are similar to those of large dimensional covariance matrices, for a large class of models studied in random matrix theory. We also derive a MarčenkoPastur type system of equations for the limiting spectral distribution of covariance matrices computed from data with elliptical distributions and generalizations of this family. The motivation for this study comes partly from the possible relevance of such distributional assumptions to problems in econometrics and portfolio optimization, as well as robustness questions for certain classical random matrix results. A mathematical theme of the paper is the important use we make of concentration inequalities.
The spectrum of kernel random matrices
, 2007
"... We place ourselves in the setting of highdimensional statistical inference, where the number of variables p in a dataset of interest is of the same order of magnitude as the number of observations n. We consider the spectrum of certain kernel random matrices, in particular n × n matrices whose (i, ..."
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Cited by 6 (2 self)
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We place ourselves in the setting of highdimensional statistical inference, where the number of variables p in a dataset of interest is of the same order of magnitude as the number of observations n. We consider the spectrum of certain kernel random matrices, in particular n × n matrices whose (i, j)th entry is f(X ′ i Xj/p) or f(‖Xi − Xj ‖ 2 /p), where p is the dimension of the data, and Xi are independent data vectors. Here f is assumed to be a locally smooth function. The study is motivated by questions arising in statistics and computer science, where these matrices are used to perform, among other things, nonlinear versions of principal component analysis. Surprisingly, we show that in highdimensions, and for the models we analyze, the problem becomes essentially linear which is at odds with heuristics sometimes used to justify the usage of these methods. The analysis also highlights certain peculiarities of models widely studied in random matrix theory and raises some questions about their relevance as tools to model highdimensional data encountered in practice. 1
ASYMPTOTIC ZERO DISTRIBUTION FOR A CLASS OF MULTIPLE ORTHOGONAL POLYNOMIALS
"... Abstract. We establish the asymptotic zero distribution for polynomials generated by a fourterm recurrence relation with varying recurrence coefficients having a particular limiting behavior. The proof is based on ratio asymptotics for these polynomials. We can apply this result to three examples o ..."
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Cited by 5 (0 self)
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Abstract. We establish the asymptotic zero distribution for polynomials generated by a fourterm recurrence relation with varying recurrence coefficients having a particular limiting behavior. The proof is based on ratio asymptotics for these polynomials. We can apply this result to three examples of multiple orthogonal polynomials, in particular JacobiPiñeiro, Laguerre I and the example associated with Macdonald functions. We also discuss an application to Toeplitz matrices. 1.
The energy of random graphs ∗
, 909
"... In 1970s, Gutman introduced the concept of the energy E (G) for a simple graph G, which is defined as the sum of the absolute values of the eigenvalues of G. This graph invariant has attracted much attention, and many lower and upper bounds have been established for some classes of graphs among whic ..."
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In 1970s, Gutman introduced the concept of the energy E (G) for a simple graph G, which is defined as the sum of the absolute values of the eigenvalues of G. This graph invariant has attracted much attention, and many lower and upper bounds have been established for some classes of graphs among which bipartite graphs are of particular interest. But there are only a few graphs attaining the equalities of those bounds. We however obtain an exact estimate of the energy for almost all graphs by Wigner’s semicircle law, which generalizes a result of Nikiforov. We further investigate the energy of random multipartite graphs by considering a generalization of Wigner matrix, and obtain some estimates of the energy for random multipartite graphs.