# The derivatives of Sinkhorn-Knopp converge

arXiv: https://arxiv.org/abs/2207.12717 · HTML: https://mlsanity.com/paper/2207.12717/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 35 references that are in the map.
- First posted on arXiv 26 Jul 2022.

## Builds on (most cited first, 13 of 13)

- [Unsupervised Learning of Visual Features by Contrasting Cluster Assignments](https://mlsanity.com/paper/2006.09882/) (2020)
- [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)
- [Learning Generative Models with Sinkhorn Divergences](https://mlsanity.com/paper/1706.00292/) (2017)
- [Optimizing Millions of Hyperparameters by Implicit Differentiation](https://mlsanity.com/paper/1911.02590/) (2019)
- [Differentiable Ranks and Sorting using Optimal Transport](https://mlsanity.com/paper/1905.11885/) (2019)
- [Optimal Transport Tools (OTT): A JAX Toolbox for all things Wasserstein](https://mlsanity.com/paper/2201.12324/) (2022)
- [Differential Properties of Sinkhorn Approximation for Learning with Wasserstein Distance](https://mlsanity.com/paper/1805.11897/) (2018)
- [A Unified Framework for Implicit Sinkhorn Differentiation](https://mlsanity.com/paper/2205.06688/) (2022)
- [Automatic differentiation of nonsmooth iterative algorithms](https://mlsanity.com/paper/2206.00457/) (2022)
- [Automatic Differentiation of Some First-Order Methods in Parametric Optimization](https://mlsanity.com/paper/1910.05696/) (2019)
- [Rethinking Initialization of the Sinkhorn Algorithm](https://mlsanity.com/paper/2206.07630/) (2022)
- [Supervised Quantile Normalization for Low-rank Matrix Approximation](https://mlsanity.com/paper/2002.03229/) (2020)

## Built on by (most cited first, 9 of 9)

- [One-step differentiation of iterative algorithms](https://mlsanity.com/paper/2305.13768/) (2023)
- [Neural Schr"odinger Bridge with Sinkhorn Losses: Application to Data-driven Minimum Effort Control of Colloidal Self-assembly](https://mlsanity.com/paper/2307.14442/) (2023)
- [Automatic Differentiation of Optimization Algorithms with Time-Varying Updates](https://mlsanity.com/paper/2410.15923/) (2024)
- [Faster Computation of Entropic Optimal Transport via Stable Low Frequency Modes](https://mlsanity.com/paper/2506.14780/) (2025)
- [Optimal Transport under Group Fairness Constraints](https://mlsanity.com/paper/2601.07144/) (2026)
- [Understanding the Curse of Unrolling](https://mlsanity.com/paper/2602.19733/) (2026)
- [Block-Wise Differentiable Sinkhorn Attention: Tail-Refinement Gradients with a Gap-Aware Dustbin Bridge](https://mlsanity.com/paper/2605.08123/) (2026)
- [Convergence of empirical subgradients for optimal transport-based objectives](https://mlsanity.com/paper/2605.28134/) (2026)
- [Support Topology and Gradient Mixing in Sinkhorn Layers](https://mlsanity.com/paper/2609.07954/) (2026)

## Abstract (arXiv)

We show that the derivatives of the Sinkhorn-Knopp algorithm, or iterative proportional fitting procedure, converge towards the derivatives of the entropic regularization of the optimal transport problem with a locally uniform linear convergence rate.