Optimal transport with $f$-divergence regularization and generalized Sinkhorn algorithm

TL;DR

This paper generalizes entropic regularization in optimal transport by incorporating all $f$-divergences of Legendre type, extending the Sinkhorn algorithm's applicability and analyzing its convergence and practical performance.

Contribution

It extends the theory of $f$-divergence regularized optimal transport to all divergences of Legendre type, providing convergence guarantees and a practical generalized Sinkhorn algorithm.

Findings

Different $f$-divergences affect convergence speed.

Choice of divergence influences numerical stability.

Regularization impacts sparsity of the optimal coupling.

Abstract

Entropic regularization provides a generalization of the original optimal transport problem. It introduces a penalty term defined by the Kullback-Leibler divergence, making the problem more tractable via the celebrated Sinkhorn algorithm. Replacing the Kullback-Leibler divergence with a general $f$-divergence leads to a natural generalization. The case of divergences defined by superlinear functions was recently studied by Di Marino and Gerolin. Using convex analysis, we extend the theory developed so far to include all $f$-divergences defined by functions of Legendre type, and prove that under some mild conditions, strong duality holds, optimums in both the primal and dual problems are attained, the generalization of the $c$-transform is well-defined, and we give sufficient conditions for the generalized Sinkhorn algorithm to converge to an optimal solution. We propose a practical algorithm for computing an approximate solution of the optimal transport problem with $f$-divergence regularization via the generalized Sinkhorn algorithm. Finally, we present experimental results on synthetic 2-dimensional data, demonstrating the effects of using different $f$-divergences for regularization, which influences convergence speed, numerical stability and sparsity of the optimal coupling.