On Unbalanced Optimal Transport: Gradient Methods, Sparsity and Approximation Error

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 $\tau$. Although only Sinkhorn-based UOT solvers have been analyzed in the literature with the complexity $O\big(\tfrac{\tau n^2\log(n)}{\varepsilon}\log\big(\tfrac{\log(n)}{{\varepsilon}}\big)\big)$ for achieving the error $\varepsilon$, their incompatibility with certain deep learning models and dense output transportation plan strongly hinder the practicality. While being vastly used as heuristics for computing UOT in modern deep learning applications and having shown success in sparse OT, gradient methods for UOT have not been formally studied. To fill this gap, we propose a novel algorithm based on Gradient Extrapolation Method (GEM-UOT) to find an $\varepsilon$-approximate solution to the UOT problem in $O\big(\kappa n^2\log\big(\frac{\tau n}{\varepsilon}\big)\big)$, where $\kappa$ is the condition number depending on the two input measures. Our algorithm is designed by optimizing a new dual formulation of the squared $\ell_2$-norm UOT objective, filling in the lack of sparse UOT literature. Finally, we establish a novel characterization of approximation error between UOT and OT in terms of both the transportation plan and transport distance. This result sheds light on a new major bottleneck neglected by the robust OT literature: though relaxing OT as UOT admits robustness to outliers, the computed UOT distance far deviates from the original OT distance. We address such limitation via a principled approach of OT retrieval from UOT based on GEM-UOT with fine tuned $\tau$ and a post-process projection step. Experiments on synthetic and real datasets validate our theories and demonstrate our methods' favorable performance.