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

arXiv: https://arxiv.org/abs/2105.14337 · HTML: https://mlsanity.com/paper/2105.14337/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 20 of its 43 references that are in the map.
- First posted on arXiv 29 May 2021.

## Builds on (most cited first, 20 of 20)

- [Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distances](https://mlsanity.com/paper/1306.0895/) (2013)
- [Computational Optimal Transport](https://mlsanity.com/paper/1803.00567/) (2018)
- [Near-linear time approximation algorithms for optimal transport via Sinkhorn iteration](https://mlsanity.com/paper/1705.09634/) (2017)
- [Stochastic Optimization for Large-scale Optimal Transport](https://mlsanity.com/paper/1605.08527/) (2016)
- [Smooth and Sparse Optimal Transport](https://mlsanity.com/paper/1710.06276/) (2017)
- [Differential Properties of Sinkhorn Approximation for Learning with Wasserstein Distance](https://mlsanity.com/paper/1805.11897/) (2018)
- [Regularized Discrete Optimal Transport](https://mlsanity.com/paper/1307.5551/) (2013)
- [An Optimal Transport approach for the Schr"odinger bridge problem and convergence of Sinkhorn algorithm](https://mlsanity.com/paper/1911.06850/) (2019)
- [A Smoothed Dual Approach for Variational Wasserstein Problems](https://mlsanity.com/paper/1503.02533/) (2015)
- [Sinkhorn Barycenters with Free Support via Frank-Wolfe Algorithm](https://mlsanity.com/paper/1905.13194/) (2019)
- [Optimal Bounds between f-Divergences and Integral Probability Metrics](https://mlsanity.com/paper/2006.05973/) (2020)
- [Regularized Optimal Transport and the Rot Mover's Distance](https://mlsanity.com/paper/1610.06447/) (2016)
- [Tsallis Regularized Optimal Transport and Ecological Inference](https://mlsanity.com/paper/1609.04495/) (2016)
- [Quadratically regularized optimal transport](https://mlsanity.com/paper/1903.01112/) (2019)
- [Generalized conditional gradient: analysis of convergence and applications](https://mlsanity.com/paper/1510.06567/) (2015)
- [Entropic regularization of continuous optimal transport problems](https://mlsanity.com/paper/1906.01333/) (2019)
- [Optimal Transport losses and Sinkhorn algorithm with general convex regularization](https://mlsanity.com/paper/2007.00976/) (2020)
- [Moreau-Yosida f-divergences](https://mlsanity.com/paper/2102.13416/) (2021)
- [Gini-regularized Optimal Transport with an Application to Spatio-Temporal Forecasting](https://mlsanity.com/paper/1712.02512/) (2017)
- [Orlicz-space regularization for optimal transport and algorithms for quadratic regularization](https://mlsanity.com/paper/1909.06082/) (2019)

## Built on by (most cited first, 4 of 4)

- [Generative Modeling through the Semi-dual Formulation of Unbalanced Optimal Transport](https://mlsanity.com/paper/2305.14777/) (2023)
- [Convergence Rates for Regularized Optimal Transport via Quantization](https://mlsanity.com/paper/2208.14391/) (2022)
- [Trust Region Policy Optimization with Optimal Transport Discrepancies: Duality and Algorithm for Continuous Actions](https://mlsanity.com/paper/2210.11137/) (2022)
- [Displacement-Sparse Neural Optimal Transport](https://mlsanity.com/paper/2502.01889/) (2025)

## Abstract (arXiv)

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.