Student Seminar Series: Kristjan Greenewald, Robust Kronecker Product PCA for Spatio-Temporal Covariance Estimation

Kronecker PCA involves the use of a space vs. time Kronecker product decomposition to estimate spatio-temporal covariances. It was shown in [G, Hero 2013] that a diagonally loaded covariance matrix is not well modeled by such a decomposition and that using a diagonal correction factor in the decomposition significantly reduces the required separation rank of the KronPCA estimate. In this work the addition of a sparse correction factor is considered, which corresponds to a model of the covariance as a sum of Kronecker products and a sparse matrix. This sparse correction includes diagonal correction as a special case but allows for sparse unstructured “outliers” anywhere in the covariance matrix. This paper introduces a robust PCA-based algorithm to estimate the covariance under this model, extending the nuclear norm penalized LS Kronecker PCA approaches of [Tsiligaridis et al 2013; G, Hero 2014]. This paper also provides an extension to Toeplitz temporal factors, 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 and compared on both simulated and real data coming from yeast cell cycle experiments. This establishes the practical utility of the sparse correction in biological and other applications.