PT$\mathrm{L}^{p}$: Partial Transport $\mathrm{L}^{p}$ Distances
This work addresses the problem of comparing signed and multi-channeled signals in machine learning, representing an incremental advancement in transport-based distance metrics.
The authors introduced partial transport L^p distances as a new family of metrics for comparing generic signals, leveraging the robustness of partial transport, and demonstrated their application in signal class separability and nearest neighbor classification.
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.