Kronecker Sum Decompositions of Space-Time Data

TL;DR

This paper explores a Kronecker sum decomposition method for estimating spatio-temporal covariance matrices, reducing sample requirements and balancing bias-variance tradeoff, demonstrated on human activity video data.

Contribution

It introduces a diagonally loaded Kronecker sum decomposition for covariance estimation, providing a smooth bias-variance tradeoff and deriving a Cramer-Rao bound for unbiased estimators.

Findings

Effective covariance estimation with fewer samples

Improved accuracy in modeling spatio-temporal data

Application to human activity video data

Abstract

In this paper we consider the use of the space vs. time Kronecker product decomposition in the estimation of covariance matrices for spatio-temporal data. This decomposition imposes lower dimensional structure on the estimated covariance matrix, thus reducing the number of samples required for estimation. To allow a smooth tradeoff between the reduction in the number of parameters (to reduce estimation variance) and the accuracy of the covariance approximation (affecting estimation bias), we introduce a diagonally loaded modification of the sum of kronecker products representation [1]. We derive a Cramer-Rao bound (CRB) on the minimum attainable mean squared predictor coefficient estimation error for unbiased estimators of Kronecker structured covariance matrices. We illustrate the accuracy of the diagonally loaded Kronecker sum decomposition by applying it to video data of human activity.