Optimal transport with $f$-divergence regularization and generalized Sinkhorn algorithm
This work provides a theoretical and algorithmic extension for researchers in optimal transport, though it is incremental as it builds on prior studies of f-divergences in this context.
The paper tackles the problem of generalizing optimal transport with entropic regularization by replacing the Kullback-Leibler divergence with any f-divergence of Legendre type, proving strong duality and convergence of a generalized Sinkhorn algorithm, and demonstrates on synthetic 2D data that different f-divergences affect convergence speed, stability, and sparsity.
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.