Sliced Wasserstein Steering between Gaussian Measures
Authors: Kaito Ito, Anqi Dong
Organizations: Department of Information Physics and Computing, The University of Tokyo, Tokyo 113-8654, Japan · Department of Decision and Control Systems and Department of Mathematics, KTH Royal Institute of Technology, Stockholm 114 28, Sweden
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.
Figures & tables
Fig. 1 : 50 samples of x(t)=[x1(t)x2(t)]⊤ under the iterative sliced controller ( 10 ) with different Td=T/h=100,1000,100000 (colored lines). The black and red ellipses are the 3σ ellipses of Σ(t) and Σf , respectively.
Fig. 2 : 2 Evolution of the 3σ ellipse of the covariance Σ(t) under the iterative sliced controller ( 10 ) with Td=100000 (black, dashed), the ideal sliced controller ( 14 ) (blue), and the minimum energy controller ( 7 ) (red). The magenta dashed ellipse shows the 3σ ellipse of Σf . 2 30 samples of x1(t) under the ideal sliced controller (in blue) and the minimum energy controller (in red).
Department of Information Physics and Computing The University of Tokyo, Tokyo 113-8654, Japan · Department of Mathematics and Digital Futures KTH Royal Institute of Technology, 100 44 Stockholm, Sweden
The Applied Mathematics and Physics Department, Graduate School of Informatics, Kyoto University, Kyoto, Japan · Daniel Guggenheim School of Aerospace, Georgia Institute of Technology, Atlanta, USA