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.

提出一种名为 COOT 的新型最优输运问题，它同时优化两个输运映射，用于异质域适应和聚类数据总结，相比现有的方法，COOT 表现出更高的性能。