Schrodinger Bridges and Density Steering Problems for Gaussian Mixtures Models in Discrete-Time

TL;DR

This paper develops a new approach for Gaussian mixture models in discrete-time Schr"{o}dinger Bridge and Density Steering problems, improving control costs and aligning with continuous-time limits.

Contribution

It constructs feasible Markovian policies as mixtures of elementary optimal policies, enhancing control cost efficiency and approximation accuracy.

Findings

Proposed policies match or improve existing control costs.

The approach aligns with recent continuous-time Schr"{o}dinger Bridge approximations.

Numerical examples validate the theoretical results.

Abstract

In this work, we revisit the discrete-time Schr\"{o}dinger Bridge (SB) and Density Steering (DS) problems for Gaussian mixture model (GMM) boundary distributions. Building on the existing literature, we construct a set of feasible Markovian policies that transport the initial distribution to the final distribution, and are expressed as mixtures of elementary component-to-component optimal policies. We then study the policy optimization within this feasible set in the context of discrete-time SBs and density-steering problems, respectively. We show that for minimum-effort density-steering problems, the proposed policy achieves the same control cost as existing approaches in the literature. For discrete-time SB problems, the proposed policy yields a cost smaller than or equal to that in the literature, resulting in a less conservative approximation. Finally, we study the continuous-time limit of our proposed discrete-time approach and show that it agrees with recently proposed approximations to the continuous-time SB for GMM boundary distributions. We illustrate this new result through two numerical examples.