Regularization of time-varying covariance matrices using linear stochastic systems

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.