Sufficient ensemble size for random matrix theory-based handling of singular covariance matrices

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Sufficient ensemble size for random matrix theory-based handling of singular covariance matrices. / Kaban, Ata.

In: Analysis and Applications, Vol. 18, No. 5, 17.07.2020, p. 929-950.

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@article{0eaac5058898441a9bbf09e28d78433e,
title = "Sufficient ensemble size for random matrix theory-based handling of singular covariance matrices",
abstract = "Singular covariance matrices are frequently encountered in both machine learning and optimization problems, most commonly due to high dimensionality of data and insufficient sample sizes. Among many methods of regularization, here we focus on a relatively recent random matrix theoretic approach, the idea of which is to create well-conditioned approximations of a singular covariance matrix and its inverse by taking the expectation of its random projections. We are interested in the error of a Monte Carlo implementation of this approach, which allows subsequent parallel processing in low dimensions in practice. We find that O(d) random projections, where d is the size of the original matrix,are sufficient for the Monte Carlo error to become negligible, in the sense of expected spectral norm difference, for both covariance and inverse covariance approximation, in the latter case under mild assumptions.",
keywords = "Singular covariance, precision matrix, curse of dimensionality, random projections;, Monte Carlo error",
author = "Ata Kaban",
year = "2020",
month = jul,
day = "17",
doi = "10.1142/S0219530520400072",
language = "English",
volume = "18",
pages = "929--950",
journal = "Analysis and Applications",
issn = "0219-5305",
publisher = "World Scientific",
number = "5",

}

RIS

TY - JOUR

T1 - Sufficient ensemble size for random matrix theory-based handling of singular covariance matrices

AU - Kaban, Ata

PY - 2020/7/17

Y1 - 2020/7/17

N2 - Singular covariance matrices are frequently encountered in both machine learning and optimization problems, most commonly due to high dimensionality of data and insufficient sample sizes. Among many methods of regularization, here we focus on a relatively recent random matrix theoretic approach, the idea of which is to create well-conditioned approximations of a singular covariance matrix and its inverse by taking the expectation of its random projections. We are interested in the error of a Monte Carlo implementation of this approach, which allows subsequent parallel processing in low dimensions in practice. We find that O(d) random projections, where d is the size of the original matrix,are sufficient for the Monte Carlo error to become negligible, in the sense of expected spectral norm difference, for both covariance and inverse covariance approximation, in the latter case under mild assumptions.

AB - Singular covariance matrices are frequently encountered in both machine learning and optimization problems, most commonly due to high dimensionality of data and insufficient sample sizes. Among many methods of regularization, here we focus on a relatively recent random matrix theoretic approach, the idea of which is to create well-conditioned approximations of a singular covariance matrix and its inverse by taking the expectation of its random projections. We are interested in the error of a Monte Carlo implementation of this approach, which allows subsequent parallel processing in low dimensions in practice. We find that O(d) random projections, where d is the size of the original matrix,are sufficient for the Monte Carlo error to become negligible, in the sense of expected spectral norm difference, for both covariance and inverse covariance approximation, in the latter case under mild assumptions.

KW - Singular covariance

KW - precision matrix

KW - curse of dimensionality

KW - random projections;

KW - Monte Carlo error

U2 - 10.1142/S0219530520400072

DO - 10.1142/S0219530520400072

M3 - Article

VL - 18

SP - 929

EP - 950

JO - Analysis and Applications

JF - Analysis and Applications

SN - 0219-5305

IS - 5

ER -