Regularized transport between singular covariance matrices

TL;DR

This paper addresses the challenge of controlling linear stochastic systems between degenerate Gaussian distributions, providing a closed-form solution for the singular covariance case and exploring its time-symmetry properties.

Contribution

It derives a feasible, closed-form interpolation for singular covariance matrices as a limit of non-degenerate cases, extending current theory to degenerate distributions.

Findings

Feasible interpolation expressed in closed form.

Interpolation belongs to the same reciprocal class as uncontrolled evolution.

Highlights time-symmetry and dual formulations of the problem.

Abstract

We consider the problem of steering a linear stochastic system between two end-point degenerate Gaussian distributions in finite time. This accounts for those situations in which some but not all of the state entries are uncertain at the initial, t = 0, and final time, t = T . This problem entails non-trivial technical challenges as the singularity of terminal state-covariance causes the control to grow unbounded at the final time T. Consequently, the entropic interpolation (Schroedinger Bridge) is provided by a diffusion process which is not finite-energy, thereby placing this case outside of most of the current theory. In this paper, we show that a feasible interpolation can be derived as a limiting case of earlier results for non-degenerate cases, and that it can be expressed in closed form. Moreover, we show that such interpolation belongs to the same reciprocal class of the uncontrolled evolution. By doing so we also highlight a time-symmetry of the problem, contrasting dual formulations in the forward and reverse time-directions, where in each the control grows unbounded as time approaches the end-point (in the forward and reverse time-direction, respectively).