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

George Rapakoulias, Fengjiao Liu, Panagiotis Tsiotras

Abstract

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.

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

Abstract

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.

Paper Structure

This paper contains 15 sections, 5 theorems, 41 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Discrete-Time Schrödinger Bridge
  4. Discrete-Time Linear Density Steering
  5. Gaussian Discrete-Time Schrödinger Bridge
  6. Gaussian Discrete-Time Linear Density Steering
  7. Schrödinger Bridge for Mixture Distributions
  8. Discrete-Time Density Steering
  9. Connections with Existing Literature
  10. Randomizing Once vs Randomizing at Every Step
  11. Continuous-Time Limit
  12. Numerical Examples
  13. Example 1
  14. Example 2
  15. Conclusions

Key Result

Proposition 1

Let $\rho_0 = \mathcal{N}(\mu_0, \Sigma_0)$, $\rho_N = \mathcal{N}(\mu_N, \Sigma_N)$ and consider a reference process $q$ associated with $x_{k+1} = x_k + \sqrt{\epsilon} \, w_k$. Then, the transition kernels and the marginal distributions of the optimal process $p^*$ that solves the dtSB problem (d where, and Furthermore, the optimal cost is given by where $V = \Sigma_N - (I_n - \epsilon N Q_0

Figures (2)

  • Figure 1: Schrodinger Bridge (left) vs Density steering (right) solutions from initial (green) Gaussian to final (blue) GMM.
  • Figure 2: Density steering double integrator dynamics from initial (green) GMM to final (blue) GMM.

Theorems & Definitions (10)

  • Proposition 1
  • proof
  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Corollary 1
  • proof
  • Theorem 3
  • proof