Unbalanced Optimal Transport meets Sliced-Wasserstein
Thibault Séjourné Clément Bonet Kilian Fatras Kimia Nadjahi Nicolas Courty
Optimal transport (OT) has emerged as a powerful framework to compare
probability measures, a fundamental task in many statistical and machine
learning problems. Substantial advances have been made over the last decade in
designing OT variants which are either computationally and statistically more
efficient, or more robust to the measures and datasets to compare. Among them,
sliced OT distances have been extensively used to mitigate optimal transport's
cubic algorithmic complexity and curse of dimensionality. In parallel,
unbalanced OT was designed to allow comparisons of more general positive
measures, while being more robust to outliers. In this paper, we propose to
combine these two concepts, namely slicing and unbalanced OT, to develop a
general framework for efficiently comparing positive measures. We propose two
new loss functions based on the idea of slicing unbalanced OT, and study their
induced topology and statistical properties. We then develop a fast
Frank-Wolfe-type algorithm to compute these loss functions, and show that the
resulting methodology is modular as it encompasses and extends prior related
work. We finally conduct an empirical analysis of our loss functions and
methodology on both synthetic and real datasets, to illustrate their relevance
and applicability.