Optimal Transport with Relaxed Marginal Constraints
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 Optimal Transport with Relaxed Marginal Constraints (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.
Authors:
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Award ID(s):
Publication Date:
NSF-PAR ID:
10233024
Journal Name:
IEEE Access
Volume:
9
Page Range or eLocation-ID:
1 to 1
ISSN:
2169-3536
2. An optimal transportation map finds the most economical way to transport one probability measure to the other. It has been applied in a broad range of applications in vision, deep learning and medical images. By Brenier theory, computing the optimal transport map is equivalent to solving a Monge-Amp\ere equation. Due to the highly non-linear nature, the computation of optimal transportation maps in large scale is very challenging. This work proposes a simple but powerful method, the FFT-OT algorithm, to tackle this difficulty based on three key ideas. First, solving Monge-Amp\ere equation is converted to a fixed point problem; Second, themore »