  We study the Unbalanced Optimal Transport (UOT) between two measures of
possibly different masses with at most $n$ components, where the marginal
constraints of 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 iteration complexity of
${O}\big(\tfrac{\tau \log(n)}{\varepsilon}
\log\big(\tfrac{\log(n)}{{\varepsilon}}\big)\big)$ and per-iteration cost of
$O(n^2)$ for achieving the desired error $\varepsilon$, their positively dense
output transportation plans strongly hinder the practicality. On the other
hand, while being vastly used as heuristics for computing UOT in modern deep
learning applications and having shown success in sparse OT problem, gradient
methods applied to UOT have not been formally studied. In this paper, we
propose a novel algorithm based on Gradient Extrapolation Method (GEM-UOT) to
find an $\varepsilon$-approximate solution to the UOT problem in $O\big( \kappa
\log\big(\frac{\tau n}{\varepsilon}\big) \big)$ iterations with
$\widetilde{O}(n^2)$ per-iteration cost, where $\kappa$ is the condition number
depending on only the two input measures. Our proof technique is based on a
novel dual formulation of the squared $\ell_2$-norm UOT objective, which fills
the lack of sparse UOT literature and also leads to a new characterization of
approximation error between UOT and OT. To this end, we further present a novel
approach of OT retrieval from UOT, which is based on GEM-UOT with fine tuned
$\tau$ and a post-process projection step. Extensive experiments on synthetic
and real datasets validate our theories and demonstrate the favorable
performance of our methods in practice.
