PT$\mathrm{L}^{p}$: Partial Transport $\mathrm{L}^{p}$ Distances
Xinran Liu Yikun Bai Huy Tran Zhanqi Zhu Matthew Thorpe Soheil Kolouri
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.