Robust Kronecker Product PCA for Spatio-Temporal Covariance Estimation

• Published in: IEEE transactions on signal processing 2015-12, Vol.63 (23), p.6368-6378
• Main Authors: Greenewald, Kristjan, Hero, Alfred O.
• Format: Article
• Language: English
• Subjects: Brain modeling; Covariance matrices; high dimensional convergence rates; Kronecker product decompositions; Linear programming; mean-square error; multivariate prediction; Optimization; Principal component analysis; robust penalized least squares; Robustness; Sparse matrices; Structured covariance estimation
• ISSN: 1053-587X; 1941-0476
• DOI: 10.1109/TSP.2015.2472364

Kronecker PCA involves the use of a space versus time Kronecker product decomposition to estimate spatio-temporal covariances. In this paper, the addition of a sparse correction factor is considered, which corresponds to a model of the covariance as a sum of Kronecker products of low (separation) rank and a sparse matrix. This sparse correction extends the diagonally corrected Kronecker PCA of [Greenewald, and Hero, 2014] to allow for sparse unstructured "outliers" anywhere in the covariance matrix, e.g., arising from variables or correlations that do not fit the Kronecker model well, or from sources such as sensor noise or sensor failure. We introduce a robust PCA-based algorithm to estimate the covariance under this model. An extension to Toeplitz temporal factors is also provided, producing a parameter reduction for temporally stationary measurement modeling. High dimensional MSE performance bounds are given for these extensions. Finally, the proposed extension of KronPCA is evaluated on both simulated and real data coming from yeast cell cycle experiments. This establishes the practical utility of robust Kronecker PCA in biological and other applications.