A hierarchically low-rank optimal transport dissimilarity measure for structured data

Abstract

We develop a class of hierarchically low-rank, scalable optimal transport dissimilarity measures for structured data, bringing the current state-of-the-art optimal transport solvers to a higher level of performance. Given two n-dimensional discrete probability measures supported on two structured grids in \({\mathbb {R}}^d\), we present a fast method for computing an entropically regularized optimal transport distance, referred to as the debiased Sinkhorn distance. The method combines Sinkhorn’s matrix scaling iteration with a low-rank hierarchical representation of the scaling matrices to achieve a near-linear complexity \({{\mathscr {O}}}(n \ln ^4 n)\). This provides a fast, scalable, and easy-to-implement algorithm for computing a class of optimal transport dissimilarity measures, enabling their applicability to large-scale optimization problems, where the computation of the classical Wasserstein metric is not feasible. We carry out a rigorous error-complexity analysis for the proposed algorithm and present several numerical examples to verify the accuracy and efficiency of the algorithm and to demonstrate its applicability in tackling real-world problems.