Schrodinger Bridges and Density Steering Problems for Gaussian Mixtures Models in Discrete-Time
This work addresses incremental improvements in optimal transport and control theory for Gaussian mixture models, relevant for applications in robotics, finance, and machine learning.
The paper tackles the discrete-time Schrödinger Bridge and Density Steering problems for Gaussian mixture models by constructing feasible Markovian policies as mixtures of component-to-component optimal policies. It shows that for density steering, the policy matches existing control costs, and for Schrödinger Bridges, it achieves a cost smaller or equal to prior work, reducing conservatism, with numerical examples validating the approach.
In this work, we revisit the discrete-time Schrödinger 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.