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The Masked Sample Covariance Estimator: An Analysis via the Matrix Laplace Transform

Chen, Richard Y. and Gittens, Alex A. and Tropp, Joel A. (2012) The Masked Sample Covariance Estimator: An Analysis via the Matrix Laplace Transform. , Pasadena, CA. (Unpublished) http://resolver.caltech.edu/CaltechAUTHORS:20120411-102106234

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Abstract

Covariance estimation becomes challenging in the regime where the number p of variables outstrips the number n of samples available to construct the estimate. One way to circumvent this problem is to assume that the covariance matrix is nearly sparse and to focus on estimating only the significant entries. To analyze this approach, Levina and Vershynin (2011) introduce a formalism called masked covariance estimation, where each entry of the sample covariance estimator is reweighed to reflect an a priori assessment of its importance. This paper provides a new analysis of the masked sample covariance estimator based on the matrix Laplace transform method. The main result applies to general subgaussian distributions. Specialized to the case of a Gaussian distribution, the theory offers qualitative improvements over earlier work. For example, the new results show that n = O(B log ^2 p) samples suffice to estimate a banded covariance matrix with bandwidth B up to a relative spectral-norm error, in contrast to the sample complexity n = O(B log ^5 p) obtained by Levina and Vershynin.


Item Type:Report or Paper (Technical Report)
Additional Information:Research supported by ONR awards N00014-08-1-0883 and N00014-11-1-0025, DARPA award N66001-08-1-2065, AFOSR award FA9550-09- 1-0643, and a Sloan Fellowship.
Group:Applied & Computational Mathematics
Funders:
Funding AgencyGrant Number
ONRN00014-08-1-0883
ONRN00014-11-1-0025
DARPAN66001-08-1-2065
AFOSRFA9550-09-1-0643
Sloan Research FellowshipUNSPECIFIED
Other Numbering System:
Other Numbering System NameOther Numbering System ID
Applied & Computational Mathematics Technical Report2012-01
Classification Code:2010 Mathematics Subject Classification: Primary: 60B20
Record Number:CaltechAUTHORS:20120411-102106234
Persistent URL:http://resolver.caltech.edu/CaltechAUTHORS:20120411-102106234
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:30057
Collection:CaltechACMTR
Deposited By: Sydney Garstang
Deposited On:30 May 2012 15:25
Last Modified:26 Dec 2012 15:03

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