Title: Transport Clustering: Solving Low-Rank Optimal Transport via Clustering
Primary area: other topics in machine learning (i.e., none of the above)
Keywords: optimal transport, low rank, approximation algorithms, k-means, co-clustering, clustering

Abstract: Optimal transport (OT) finds a least cost transport plan between two probability distributions using a cost matrix over pairs of points. Constraining the rank of the transport plan yields low-rank OT, which improves computational complexity and statistical stability compared to full-rank OT.  Further, low-rank OT naturally induces co-clusters between distributions and generalizes $K$-means clustering. Reversing this direction, we show that solving a clustering problem on a set of _correspondences_, termed _transport clustering_, solves low-rank OT. This connection between low-rank OT and transport clustering relies on a _transport registration_ of the cost matrix which registers the cost matrix via the transport map. We show that the reduction of low-rank OT to transport clustering yields polynomial-time, constant-factor approximation algorithms for low-rank OT. Specifically, we show that for the low-rank OT problem this reduction yields a $(1+\gamma)$-approximation algorithm for metrics of negative-type and a $(1+\gamma+\sqrt{2\gamma}\,)$-approximation algorithm for kernel costs where $\gamma \in [0,1]$ denotes the approximation ratio to the optimal full-rank solution. We demonstrate that transport clustering outperforms existing low-rank OT methods on several synthetic benchmarks and large-scale, high-dimensional real datasets.