Eigen structure of a new class of structured covariance and inverse covariance matrices

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Title: Eigen structure of a new class of structured covariance and inverse covariance matrices
Author(s): Battey, HS
Item Type: Journal Article
Abstract: There is a one to one mapping between a p dimensional strictly positive definite covariance matrix Σ and its matrix logarithm L. We exploit this relationship to study the structure induced on Σ through a sparsity constraint on L. Consider L as a random matrix generated through a basis expansion, with the support of the basis coefficients taken as a simple random sample of size s = s ∗ from the index set [p(p + 1)/2] = {1, . . . , p(p + 1)/2}. We find that the expected number of non-unit eigenvalues of Σ, denoted E[|A|], is approximated with near perfect accuracy by the solution of the equation 4p + p(p − 1) 2(p + 1) h log p p − d − d 2p(p − d) i − s ∗ = 0. Furthermore, the corresponding eigenvectors are shown to possess only p − |Ac | nonzero entries. We use this result to elucidate the precise structure induced on Σ and Σ−1 . We demonstrate that a positive definite symmetric matrix whose matrix logarithm is sparse is significantly less sparse in the original domain. This finding has important implications in high dimensional statistics where it is important to exploit structure in order to construct consistent estimators in non-trivial norms. An estimator exploiting the structure of the proposed class is presented.
Publication Date: 23-May-2017
Date of Acceptance: 8-Mar-2016
URI: http://hdl.handle.net/10044/1/37510
DOI: https://dx.doi.org/10.3150/16-BEJ840
ISSN: 1350-7265
Publisher: Bernoulli Society for Mathematical Statistics and Probability
Start Page: 3166
End Page: 3177
Journal / Book Title: Bernoulli
Volume: 23
Issue: 4B
Copyright Statement: © 2017 ISI/BS
Keywords: 0104 Statistics
1403 Econometrics
Statistics & Probability
Publication Status: Published
Appears in Collections:Mathematics
Faculty of Natural Sciences

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