On the Convergence of Gradient Extrapolation Methods for Unbalanced Optimal Transport

by   Quang Minh Nguyen, et al.

We study the Unbalanced Optimal Transport (UOT) between two measures of possibly different masses with at most n components, where marginal constraints of the standard Optimal Transport (OT) are relaxed via Kullback-Leibler divergence with regularization factor τ. We propose a novel algorithm based on Gradient Extrapolation Method (GEM-UOT) to find an ε-approximate solution to the UOT problem in O( κ n^2 log(τ n/ε) ), where κ is the condition number depending on only the two input measures. Compared to the only known complexity O(τ n^2 log(n)εlog(log(n)ε)) for solving the UOT problem via the Sinkhorn algorithm, ours is better in ε and lifts Sinkhorn's linear dependence on τ, which hindered its practicality to approximate the standard OT via UOT. Our proof technique is based on a novel dual formulation of the squared ℓ_2-norm regularized UOT objective, which is of independent interest and also leads to a new characterization of approximation error between UOT and OT in terms of both the transportation plan and transport distance. To this end, we further present an algorithm, based on GEM-UOT with fine tuned τ and a post-process projection step, to find an ε-approximate solution to the standard OT problem in O( κ n^2 log( n/ε) ), which is a new complexity in the literature of OT. Extensive experiments on synthetic and real datasets validate our theories and demonstrate the favorable performance of our methods in practice.


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