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

TL;DR

The paper proposes a unifying, interpretable distance between probability laws based on the discrepancy between the CDFs of one-dimensional projections, $\Delta(X,Y) = \mathbb{E}_{q}[ d(F_X^{q}, F_Y^{q}) ]$, enabling scalable comparison of high-dimensional distributions. A practical, differentiable estimator $\hat{\Delta}(X,Y)$ is developed with asymptotic guarantees, and a concrete algorithm computes this distance efficiently, facilitating gradient-based control. The authors demonstrate the approach on two control tasks: distribution steering and ergodic control, using gradient descent (and AdamW) to steer state distributions and to align trajectory distributions with a target, respectively. The framework generalizes existing sliced distance concepts and supports variants that recover sliced KS or sliced 1-Wasserstein, providing a versatile tool for distributional analysis in control. Overall, the method offers a tractable, differentiable means to quantify and minimize distributional discrepancy in real-time control settings, with clear avenues for future theoretical and algorithmic refinement.

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.

Figures (4)

• Figure 1: Overview of our proposed family of distances between the distributions of two random variables $X$ and $Y$. We average discrepancies between the CDFs of linear one-dimensional projections of the random variables. The distributions of the random variables are represented as sets of samples here.
• Figure 2: The empirical distribution of $D_{300, 100}(X,Y)$ for the null case and different alternative cases in the two-sample test. The randomness comes from the drawn samples for $X$ and $Y$ and the selection of half-spaces in \ref{['alg:dist_heur']}. In all experiments, the null case is $\mu_X=\mu_Y=\mathcal{N}(0,I)$. Each experiment corresponds to a particular choice of the dimensionality $n$ and the alternative case. The distribution of $D_{300, 100}(X,Y)$ is given in blue for the null case and in orange for the alternative. The mean values are denoted by the vertical dotted lines.
• Figure 3: Results for the distribution steering application.
• Figure 4: Results for the ergodic control application.

Theorems & Definitions (12)

• Definition 1
• Definition 2
• Definition 3
• Theorem 1
• proof
• Definition 4
• Theorem 2
• proof
• Corollary 1
• proof