Kavli Affiliate: Yi Zhou
| First 5 Authors: Quang Minh Nguyen, Hoang H. Nguyen, Yi Zhou, Lam M. Nguyen,
| Summary:
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 $Obig(tfrac{tau
n^2log(n)}{varepsilon}logbig(tfrac{log(n)}{{varepsilon}}big)big)$ 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 $Obig(kappa
n^2logbig(frac{tau n}{varepsilon}big)big)$, 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.
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