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.

翻译：最优传输及其相关问题（包括最优部分传输）已被证明是机器学习中用于计算概率测度或正测度间有意义距离的有效工具。这一成功使得人们对定义基于传输的距离度量产生了日益浓厚的兴趣，以实现有符号测度乃至更一般的多通道信号间的比较。传输$\mathrm{L}^{p}$距离是将最优传输框架扩展至有符号信号及可能的多通道信号的重要延伸。本文提出了一类新的度量——部分传输$\mathrm{L}^{p}$距离，用于比较一般信号，该度量兼具部分传输距离的鲁棒性优势。我们提供了理论基础，包括最优方案的存在性以及该距离在各种极限情形下的行为特征。此外，我们还引入了这些距离的切片变体，可实现对一般信号的快速比较。最后，我们展示了所提出的距离在信号类可分性及最近邻分类中的应用。