Intrinsically Bayesian robust Karhunen-Loève compression

Abstract

Karhunen-Loève (KL) compression is based on the canonical representation of a random process. When compressing to a finite sum, the optimal-MSE m-term summation consists of the KL terms possessing the m largest eigenvalues. This paper considers the situation in which an unknown covariance matrix belongs to an uncertainty class governed by a prior probability distribution. The intrinsically Bayesian robust (IBR) KL compression minimizes the expected MSE over the uncertainty class among all possible m-term KL expansions. We prove that the IBR KL compression is the KL expansion based on the expected covariance matrix over the uncertainty class. We then solve the following experimental design problem: among the unknown covariances, which should be determined to maximally reduce the mean objective cost of uncertainty (MOCU), which in the KL compression setting measures increased MSE resulting from our uncertainty. The IBR KL expansion and optimal experimental design are solved analytically for the Wishart distribution over the uncertainty class, a commonly employed distribution for covariance matrices in Bayesian settings.