Robust Kronecker Product PCA for Spatio-Temporal Covariance Estimation

TL;DR

This paper introduces a robust Kronecker PCA method that incorporates a sparse correction factor for improved spatio-temporal covariance estimation, especially in the presence of outliers or noise, with applications demonstrated in biological data.

Contribution

It extends existing Kronecker PCA models by adding a sparse correction factor and develops a robust estimation algorithm, including Toeplitz temporal extensions, with theoretical performance bounds.

Findings

Effective in handling outliers and sensor noise.

Improved covariance estimation in high dimensions.

Validated on biological yeast cell cycle data.

Abstract

Kronecker PCA involves the use of a space vs. time Kronecker product decomposition to estimate spatio-temporal covariances. 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 of low (separation) rank and a sparse matrix. This sparse correction extends the diagonally corrected Kronecker PCA of [Greenewald et al 2013, 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, extending the rearranged nuclear norm penalized LS Kronecker PCA approaches of [Greenewald et al 2014, Tsiligkaridis et al 2013]. 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.