Strong convergence of the empirical distribution of eigenvalues of sample covariance matrices with a perturbation matrix

Pan, Guangming

doi:10.1016/j.jmva.2010.02.001

Cited by 20 publications

(27 citation statements)

References 9 publications

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“…are random subspace coefficients and have mean zero and unit variance, and • ε i ∈ R d are independent noise vectors that have entries ε ij iid ∼ F 3 (0, 1) with mean zero and unit variance, such that the distributions F 1 and F 2 satisfy the log-Sobolev inequality [13] and the distribution F 3 satisfies condition (1.3) from [14]. Notably, these conditions are satisfied by…”

Section: Resultsmentioning

confidence: 99%

“…and P is generated according to the "i.i.d model" and satisfies Assumption 2.4 of [12], and X satisfies Assumptions 2.1-2.3 of [12] (X here matches the random matrix in [14], which [12] refers to as an example of a random matrix that satisfies the assumptions). Thus under the condition ϕ (b + ) = −∞ (we will show in subsection IV-C that it is indeed satisfied), Theorems 2.10 and 2.11 from [12] yield…”

Section: A Obtain An Initial Expressionmentioning

confidence: 99%

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Towards a theoretical analysis of PCA for heteroscedastic data

Hong

Balzano

Fessler

2016

2016 54th Annual Allerton Conference on Communication, Control, and Computing (Allerton)

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Abstract-Principal Component Analysis (PCA) is a method for estimating a subspace given noisy samples. It is useful in a variety of problems ranging from dimensionality reduction to anomaly detection and the visualization of high dimensional data. PCA performs well in the presence of moderate noise and even with missing data, but is also sensitive to outliers. PCA is also known to have a phase transition when noise is independent and identically distributed; recovery of the subspace sharply declines at a threshold noise variance. Effective use of PCA requires a rigorous understanding of these behaviors. This paper provides a step towards an analysis of PCA for samples with heteroscedastic noise, that is, samples that have nonuniform noise variances and so are no longer identically distributed. In particular, we provide a simple asymptotic prediction of the recovery of a one-dimensional subspace from noisy heteroscedastic samples. The prediction enables: a) easy and efficient calculation of the asymptotic performance, and b) qualitative reasoning to understand how PCA is impacted by heteroscedasticity (such as outliers).

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Section: Resultsmentioning

confidence: 99%

Section: A Obtain An Initial Expressionmentioning

confidence: 99%

Towards a theoretical analysis of PCA for heteroscedastic data

Hong

Balzano

Fessler

2016

2016 54th Annual Allerton Conference on Communication, Control, and Computing (Allerton)

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show abstract

“…The LSD of S 2t and its Stieltjes transform are denoted by F y 2 t and m y 2 t (z), respectively. Under Assumptions 2-4, from [17] and [16], m y 2 t (z) is the unique solution in C + to…”

Section: Methodology and Theorymentioning

confidence: 99%

Independence test for high dimensional data based on regularized canonical correlation coefficients

Yang¹,

Pan²

2015

Ann. Statist.

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This paper proposes a new statistic to test independence between two high dimensional random vectors X : p 1 × 1 and Y : p 2 × 1. The proposed statistic is based on the sum of regularized sample canonical correlation coefficients of X and Y. The asymptotic distribution of the statistic under the null hypothesis is established as a corollary of general central limit theorems (CLT) for the linear statistics of classical and regularized sample canonical correlation coefficients when p 1 and p 2 are both comparable to the sample size n. As applications of the developed independence test, various types of dependent structures, such as factor models, ARCH models and a general uncorrelated but dependent case, etc., are investigated by simulations. As an empirical application, cross-sectional dependence of daily stock returns of companies between different sections in the New York Stock Exchange (NYSE) is detected by the proposed test.

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“…The limiting spectral distributions of F-matrices were studied in Wachter [1980], Bai et al [1987]. In addition, products of random matrices arise in study of high dimensional time-series, for example, see Pan [2010], Pan and Gao [2012]. For a history of the product of random matrices the reader is referred to Bai et al [2007].…”

Section: Introductionmentioning

confidence: 99%

Regular variation and free regular infinitely divisible laws

Chakrabarty

Chakraborty

Hazra

2020

Statistics & Probability Letters

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In this article the relation between the tail behaviours of a free regular infinitely divisible probability measure and its Lévy measure is studied. An important example of such a measure is the compound free Poisson distribution, which often occurs as a limiting spectral distribution of certain sequences of random matrices. We also describe a connection between an analogous classical result of Embrechts et al. [1979] and our result using the Bercovici-Pata bijection.2010 Mathematics Subject Classification. 60G70; 46L53; 46L54.

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Strong convergence of the empirical distribution of eigenvalues of sample covariance matrices with a perturbation matrix

Cited by 20 publications

References 9 publications

Towards a theoretical analysis of PCA for heteroscedastic data

Towards a theoretical analysis of PCA for heteroscedastic data

Independence test for high dimensional data based on regularized canonical correlation coefficients

Regular variation and free regular infinitely divisible laws

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