通过切片投影实现高维概率分布的低耗能精准调控
Sliced Wasserstein Steering between Gaussian Measures
- 将高维分布投影到球面上各一维方向，分方向求解最优控制
- 在高斯分布下可精确追踪目标分布，能耗与切片Wasserstein距离相关
- 适合传感器仅观测部分线性信息的机器人控制场景
二次代价最优传输为以最小能量操控概率分布提供了几何框架。然而，高维环境下传统空间方法变得复杂，实际感知或执行常仅提供状态的线性视图——如相机轮廓、激光雷达束、断层扫描切片。本文提出一种切片反馈控制器：将演化分布投影至球面上的一维方向，在每个投影中合成最优一维速度，并平均这些速度生成环境空间中的反馈控制。该构造退化为一维的Benamou-Brenier问题。此外，其在正交变换下不变，投影下非扩张，且在$ mathcal{P}_2( mathbb{R}^n)$上良定。计算通过在球面上采样方向并独立求解一维子问题实现，具有可扩展性，契合部分观测条件。在高斯设定下，证明所提出的切片控制器可将分布精确导向目标分布。进一步推导出控制器能耗与切片Wasserstein距离之间的恒等关系。
原文摘要 · Abstract (English)
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.
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