On metrics robust to noise and deformations

TL;DR

This paper investigates a family of integral probability metrics for comparing functions, demonstrating their robustness to noise and deformations, and providing error bounds for finite sample approximations, with practical experiments included.

Contribution

It introduces and analyzes the robustness properties of a class of metrics for signals, including error bounds and empirical validation, extending their applicability to noisy and deformed data.

Findings

Metrics are robust to broad classes of deformations.

Finite sample approximations are robust to Gaussian noise.

Numerical experiments compare these metrics with Wasserstein distances.

Abstract

We study the properties of a family of distances between functions of a single variable. These distances are examples of integral probability metrics, and have been used previously for comparing probability measures on the line; special cases include the Earth Mover's Distance and the Kolmogorov Metric. We examine their properties for general signals, proving that they are robust to a broad class of deformations. We also establish corresponding robustness results for the induced sliced distances between multivariate functions. Finally, we establish error bounds for approximating the univariate metrics from finite samples, and prove that these approximations are robust to additive Gaussian noise. The results are illustrated in numerical experiments, which include comparisons with Wasserstein distances.