Abstract
Optimal transport (OT) is a principled approach for matching, having achieved success in diverse applications such as tracking and cluster alignment. It is also the core computation problem for solving the Wasserstein metric between probabilistic distributions, which has been increasingly used in machine learning. Despite its popularity, the marginal constraints of OT impose fundamental limitations. For some matching or pattern extraction problems, the framework of OT is not suitable, and post-processing of the OT solution is often unsatisfactory. In this paper, we extend OT by a new optimization formulation called <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Optimal Transport with Relaxed Marginal Constraints</i> (OT-RMC). Specifically, we relax the marginal constraints by introducing a penalty on the deviation from the constraints. Connections with the standard OT are revealed both theoretically and experimentally. We demonstrate how OT-RMC can easily adapt to various tasks by three highly different applications in image analysis and single-cell data analysis. Quantitative comparisons have been made with OT and another commonly used matching scheme to show the remarkable advantages of OT-RMC.
Highlights
Optimal transport (OT) has been successfully applied in diverse areas including machine learning, computer vision, and bioinformatics
EXPERIMENTS We illustrate the usage of Optimal Transport with Relaxed Marginal Constraints (OT-RMC) with three example applications
To show the pixels that are selected by OT-RMC, meaning that the representative colors of the clusters which the pixels belong to are selected, we show these pixels in the original color, while the pixels not selected are shown in brightened gray scale
Summary
Optimal transport (OT) has been successfully applied in diverse areas including machine learning, computer vision, and bioinformatics. When the distributions are finite and discrete, Wasserstein distance is solved by linear programming In this case, it will become clear that OT fits well with our intuition of optimal matching. When OT is used to match (aka align) clusters in different clustering results and subsequently to consolidate clusters in those results, the marginal constraints imply that the proportions of the true clusters are fixed across the results. In another word, it is implicitly assumed that the variation observed in the results arises from the randomness in the data or other nuance factors.
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