Optimal transport (OT) has gained popularity due to its various applications in fields such as machine learning, statistics, and signal processing. However, the balanced mass requirement limits its performance in practical problems. To address these limitations, variants of the OT problem, including unbalanced OT, Optimal partial transport (OPT), and Hellinger Kantorovich (HK), have been proposed. In this paper, we propose the Linear optimal partial transport (LOPT) embedding, which extends the (local) linearization technique on OT and HK to the OPT problem. The proposed embedding allows for faster computation of OPT distance between pairs of positive measures. Besides our theoretical contributions, we demonstrate the LOPT embedding technique in point-cloud interpolation and PCA analysis.

翻译：最优传输（OT）因其在机器学习、统计学和信号处理等领域的广泛应用而受到关注。然而，等质量要求限制了其在实际问题中的性能。为解决这些限制，研究者提出了OT问题的变体，包括非平衡OT、最优部分传输（OPT）和Hellinger Kantorovich（HK）。本文提出线性最优部分传输（LOPT）嵌入，将OT和HK上的（局部）线性化技术扩展到OPT问题。所提出的嵌入能够更快地计算正测度对之间的OPT距离。除理论贡献外，我们还在点云插值和PCA分析中展示了LOPT嵌入技术的应用。