PT$\mathrm{L}^{p}$: Partial Transport $\mathrm{L}^{p}$ Distances

TL;DR

This paper introduces partial transport L^p distances, a new family of metrics for comparing signals that extend optimal transport to signed and multi-channeled signals, with theoretical foundations and practical applications in signal classification.

Contribution

The paper proposes partial transport L^p distances as a novel metric for signals, including theoretical analysis and a sliced variation for efficient comparison.

Findings

Theoretical existence of optimal plans for the new distances.

The sliced variation enables rapid signal comparison.

Application in signal classification improves separability.

Abstract

Optimal transport and its related problems, including optimal partial transport, have proven to be valuable tools in machine learning for computing meaningful distances between probability or positive measures. This success has led to a growing interest in defining transport-based distances that allow for comparing signed measures and, more generally, multi-channeled signals. Transport $\mathrm{L}^{p}$ distances are notable extensions of the optimal transport framework to signed and possibly multi-channeled signals. In this paper, we introduce partial transport $\mathrm{L}^{p}$ distances as a new family of metrics for comparing generic signals, benefiting from the robustness of partial transport distances. We provide theoretical background such as the existence of optimal plans and the behavior of the distance in various limits. Furthermore, we introduce the sliced variation of these distances, which allows for rapid comparison of generic signals. Finally, we demonstrate the application of the proposed distances in signal class separability and nearest neighbor classification.