# Sliced Wasserstein Steering between Gaussian Measures

arXiv: https://arxiv.org/abs/2604.22807 · HTML: https://mlsanity.com/paper/2604.22807/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 9 of its 28 references that are in the map.
- First posted on arXiv 14 Apr 2026.

## Builds on (most cited first, 9 of 9)

- [On the relation between optimal transport and Schr"odinger bridges: A stochastic control viewpoint](https://mlsanity.com/paper/1412.4430/) (2014)
- [Sliced-Wasserstein Flows: Nonparametric Generative Modeling via Optimal Transport and Diffusions](https://mlsanity.com/paper/1806.08141/) (2018)
- [Optimal transport for Gaussian mixture models](https://mlsanity.com/paper/1710.07876/) (2017)
- [Optimal transport over a linear dynamical system](https://mlsanity.com/paper/1502.01265/) (2015)
- [Long-Time Asymptotics of the Sliced-Wasserstein Flow](https://mlsanity.com/paper/2405.06313/) (2024)
- [Monge-Kantorovich Optimal Transport Through Constrictions and Flow-rate Constraints](https://mlsanity.com/paper/2212.14509/) (2022)
- [Entropic Model Predictive Optimal Transport over Dynamical Systems](https://mlsanity.com/paper/2301.06492/) (2023)
- [Towards Understanding Gradient Dynamics of the Sliced-Wasserstein Distance via Critical Point Analysis](https://mlsanity.com/paper/2502.06525/) (2025)
- [Entropic Model Predictive Optimal Transport for Underactuated Linear Systems](https://mlsanity.com/paper/2308.07599/) (2023)

## Built on by (most cited first, 1 of 1)

- [Distribution Steering via Sliced Optimal Transport Control](https://mlsanity.com/paper/2608.12828/) (2026)

## Abstract (arXiv)

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 P2(ℝn). 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.