Regularization of time-varying covariance matrices using linear stochastic systems

TL;DR

This paper models time-varying covariance matrices through linear stochastic systems, introducing new covariance paths based on regularizations inspired by optimal transport and Schrödinger bridge concepts, with explicit formulas and comparisons.

Contribution

It develops differential equations and closed-form expressions for novel covariance paths induced by regularizations on linear system matrices.

Findings

Derived differential equations for covariance paths.

Provided closed-form solutions for scalar covariance.

Compared different covariance paths through examples.

Abstract

This work focuses on modeling of time-varying covariance matrices using the state covariance of linear stochastic systems. Following concepts from optimal mass transport and the Schr\"odinger bridge problem (SBP), we investigate several covariance paths induced by different regularizations on the system matrices. Specifically, one of the proposed covariance path generalizes the geodesics based on the Fisher-Rao metric to the situation with stochastic input. Another type of covariance path is generated by linear system matrices with rotating eigenspace in the noiseless situation. The main contributions of this paper include the differential equations for these covariance paths and the closed-form expressions for scalar-valued covariance. We also compare these covariance paths using several examples.