# Optimal Transport losses and Sinkhorn algorithm with general convex regularization

arXiv: https://arxiv.org/abs/2007.00976 · HTML: https://mlsanity.com/paper/2007.00976/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 2 Jul 2020.

## Built on by (most cited first, 10 of 10)

- [Convergence Rates for Regularized Optimal Transport via Quantization](https://mlsanity.com/paper/2208.14391/) (2022)
- [Efficient and Accurate Optimal Transport with Mirror Descent and Conjugate Gradients](https://mlsanity.com/paper/2307.08507/) (2023)
- [Optimal transport with f-divergence regularization and generalized Sinkhorn algorithm](https://mlsanity.com/paper/2105.14337/) (2021)
- [Bringing regularized optimal transport to lightspeed: a splitting method adapted for GPUs](https://mlsanity.com/paper/2305.18483/) (2023)
- [Estimating 2-Sinkhorn Divergence between Gaussian Processes from Finite-Dimensional Marginals](https://mlsanity.com/paper/2102.03267/) (2021)
- [UTOPIA: Unconstrained Tracking Objects without Preliminary Examination via Cross-Domain Adaptation](https://mlsanity.com/paper/2306.09613/) (2023)
- [NTIRE 2025 Image Shadow Removal Challenge Report](https://mlsanity.com/paper/2506.15524/) (2025)
- [Convergence rate of Tsallis entropic regularized optimal transport](https://mlsanity.com/paper/2304.06616/) (2023)
- [Feature-aligned N-BEATS with Sinkhorn divergence](https://mlsanity.com/paper/2305.15196/) (2023)
- [Freeze, Diffuse, Decode: Task-Aware Adaptation of Transformer Embeddings for Antimicrobial Peptide Design](https://mlsanity.com/paper/2511.23120/) (2025)

## Abstract (arXiv)

We introduce a new class of convex-regularized Optimal Transport losses, which generalizes the classical Entropy-regularization of Optimal Transport and Sinkhorn divergences, and propose a generalized Sinkhorn algorithm. Our framework unifies many regularizations and numerical methods previously appeared in the literature. We show the existence of the maximizer for the dual problem, complementary slackness conditions, providing a complete characterization of solutions for such class of variational problems. As a consequence, we study structural properties of these losses, including continuity, differentiability and provide explicit formulas for the its gradient. Finally, we provide theoretical guarantees of convergences and stability of the generalized Sinkhorn algorithm, even in the continuous setting. The techniques developed here are directly applicable also to study Wasserstein barycenters or, more generally, multi-marginal problems.