Hierarchical Refinement: Optimal Transport to Infinity and Beyond

Optimal transport (OT) has enjoyed great success in machine-learning as a principled way to align datasets via a least-cost correspondence. This success was driven in large part by the runtime efficiency of the Sinkhorn algorithm [Cuturi 2013], which computes a coupling between points from two datasets. However, Sinkhorn has quadratic space complexity in the number of points, limiting the scalability to larger datasets. Low-rank OT achieves linear-space complexity, but by definition, cannot compute a one-to-one correspondence between points. When the optimal transport problem is an assignment problem between datasets then the optimal mapping, known as the Monge map, is guaranteed to be a bijection. In this setting, we show that the factors of an optimal low-rank coupling co-cluster each point with its image under the Monge map. We leverage this invariant to derive an algorithm, Hierarchical Refinement (HiRef), that dynamically constructs a multiscale partition of a dataset using low-rank OT subproblems, culminating in a bijective coupling. Hierarchical Refinement uses linear space and has log-linear runtime, retaining the space advantage of low-rank OT while overcoming its limited resolution. We demonstrate the advantages of Hierarchical Refinement on several datasets, including ones containing over a million points, scaling full-rank OT to problems previously beyond Sinkhorn's reach.