Sliced Wasserstein Steering between Gaussian Measures

TL;DR

This paper introduces a sliced Wasserstein steering method for probability distributions that reduces high-dimensional optimal transport problems to manageable one-dimensional problems, enabling scalable and observation-aligned control.

Contribution

It develops a novel sliced feedback controller for distribution steering that is invariant, nonexpansive, and scalable, especially effective with partial observations and Gaussian measures.

Findings

The sliced controller successfully steers Gaussian laws to targets.

The method is invariant under orthogonal transformations.

Energy consumption relates directly to sliced Wasserstein distance.

Abstract

Optimal transport with quadratic cost provides a geometric framework for steering an ensemble, modeled by a probability law, with minimal effort. Yet ambient-space formulations become unwieldy in high dimensions, and sensing or actuation in practice often reveals only linear views of the state -- camera silhouettes, LiDAR beams, tomographic slices. We develop a sliced feedback controller for distribution steering: the evolving law is projected onto one-dimensional directions on the sphere, the optimal one-dimensional velocity is synthesized in each projection, and these velocities are averaged to produce a feedback control in the ambient space. The construction reduces to the Benamou--Brenier problem in one dimension. In addition, it is invariant under orthogonal transforms, nonexpansive under projections, and well posed on $\mathcal{P}_2(\mathbb{R}^n)$. Computation proceeds by sampling directions on the sphere and solving independent one-dimensional subproblems, yielding a scalable method aligned with partial observations. In the Gaussian setting, we show that the developed sliced controller steers the law to the prescribed target. Furthermore, we derive an identity relating the energy consumption incurred by the controller to the sliced Wasserstein distance.