Linear Optimal Partial Transport Embedding

TL;DR

This paper introduces the Linear optimal partial transport (LOPT) embedding, a novel method that extends linearization techniques to unbalanced optimal transport problems, enabling faster computations and applications in point-cloud interpolation and PCA.

Contribution

The paper presents the LOPT embedding, a new approach that extends linearization to unbalanced OT, improving computational efficiency and broadening application scope.

Findings

Enables faster computation of OPT distances.

Demonstrates effectiveness in point-cloud interpolation.

Shows utility in PCA analysis.

Abstract

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.