The LQR-Schr{\"o}dinger Bridge

TL;DR

This paper introduces a closed-form solution for the LQR-Schr{"o}dinger bridge problem in discrete time with Gaussian marginals, leveraging Riccati equations to construct complex Gaussian processes and extend Bures transport.

Contribution

It provides a novel closed-form solution for the LQR-Schr{"o}dinger bridge with Gaussian marginals using Riccati equations, enabling complex process construction and geometric extensions.

Findings

Closed-form solution for Gaussian LQR-Schr{"o}dinger bridge

Efficient convergence of Riccati-based dual system

Ability to construct complex Gaussian processes with desired properties

Abstract

We consider the Schr{\"o}dinger bridge problem in discrete time, where the pathwise cost is replaced by a sum of quadratic functions, taking the form of a linear quadratic regulator (LQR) cost. This cost comprises potential terms that act as attractors and kinetic terms that control the diffusion of the process. When the two boundary marginals are Gaussian, we show that the LQR-Schr{\"o}dinger bridge problem can be solved in closed form. We follow the dynamic programming principle, interpreting the Kantorovich potentials as cost-to-go functions. Under the LQR-Gaussian assumption, these potentials can be propagated exactly in a backward and forward passes, leading to a system of dual Riccati equations, well known in estimation and control. This system converges rapidly in practice. We then show that the optimal process is Markovian and compute its transition kernel in closed form as well as the Gaussian marginals. Through numerical experiments, we demonstrate that this approach can be used to construct complex, non-homogeneous Gaussian processes with acceleration and loops, given well-chosen attractive potentials. Moreover, this approach allows extending the Bures transport between Gaussian distributions to more complex geometries with negative curvature.