Regularized block Toeplitz covariance matrix estimation via Kronecker product expansions

Abstract

In this work we consider the estimation of spatio-temporal covariance matrices in the low sample non-Gaussian regime. We impose covariance structure in the form of a sum of Kronecker products decomposition [1, 2] with diagonal correction [2], which we refer to as DC-KronPCA, in the estimation of multiframe covariance matrices. This paper extends the approaches of [1] in two directions. First, we modify the diagonally corrected method of [2] to include a block Toeplitz constraint imposing temporal stationarity structure. Second, we improve the conditioning of the estimate in the very low sample regime by using Ledoit-Wolf type shrinkage regular-ization similar to [3]. For improved robustness to heavy tailed distributions, we modify the KronPCA to incorporate robust shrinkage estimation [4]. Results of numerical simulations establish benefits in terms of estimation MSE when compared to previous methods. Finally, we apply our methods to a real-world network spatio-temporal anomaly detection problem and achieve superior results.

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