CO-Optimal Transport
This addresses a limitation in optimal transport for machine learning applications where data are from heterogeneous domains, offering a novel solution with practical benefits.
The paper tackles the problem of applying optimal transport to distributions on different spaces by proposing CO-Optimal Transport (COOT), which simultaneously optimizes transport maps for samples and features, leading to performance improvements in heterogeneous domain adaptation and co-clustering.
Optimal transport (OT) is a powerful geometric and probabilistic tool for finding correspondences and measuring similarity between two distributions. Yet, its original formulation relies on the existence of a cost function between the samples of the two distributions, which makes it impractical when they are supported on different spaces. To circumvent this limitation, we propose a novel OT problem, named COOT for CO-Optimal Transport, that simultaneously optimizes two transport maps between both samples and features, contrary to other approaches that either discard the individual features by focusing on pairwise distances between samples or need to model explicitly the relations between them. We provide a thorough theoretical analysis of our problem, establish its rich connections with other OT-based distances and demonstrate its versatility with two machine learning applications in heterogeneous domain adaptation and co-clustering/data summarization, where COOT leads to performance improvements over the state-of-the-art methods.