Sliced Distribution Matching based on Cumulative Distribution Functions with Applications to Control

TL;DR

This paper introduces a new family of distance measures between probability distributions based on cumulative distribution functions of projections, with applications in control, offering interpretability, simplicity, and differentiability.

Contribution

A unified, interpretable distance based on CDF discrepancies of projections, with theoretical guarantees and practical applications in control and distribution testing.

Findings

Effective in two-sample testing to distinguish distributions

Enables gradient-based control solutions

Provides asymptotic guarantees for estimators

Abstract

Computing the similarity between two probability distributions is a recurring theme across control. We introduce a unified family of distances between the probability distributions of two random variables that is based on the discrepancy between the cumulative distribution functions of random linear one-dimensional projections of the random variables. Our proposed distance is interpretable, computationally simple, and admits a differentiable approximation. We establish asymptotic theoretical guarantees for sample-based estimators of the distance. We empirically study the use of the distance in a two-sample test and demonstrate its ability to distinguish different distributions. Finally, we show that the distance allows for simple gradient-based solutions in control by studying distribution steering and ergodic control.