# Displacement smoothness of entropic optimal transport

arXiv: https://arxiv.org/abs/2210.00225 · HTML: https://mlsanity.com/paper/2210.00225/

## Key facts

- Cited by 13 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 1 Oct 2022.

## Built on by (most cited first, 13 of 13)

- [One-Step Generative Modeling via Wasserstein Gradient Flows](https://mlsanity.com/paper/2605.11755/) (2026)
- [Doubly Regularized Entropic Wasserstein Barycenters](https://mlsanity.com/paper/2303.11844/) (2023)
- [Progressive Entropic Optimal Transport Solvers](https://mlsanity.com/paper/2406.05061/) (2024)
- [Estimating the Rate-Distortion Function by Wasserstein Gradient Descent](https://mlsanity.com/paper/2310.18908/) (2023)
- [A semiconcavity approach to stability of entropic plans and exponential convergence of Sinkhorn's algorithm](https://mlsanity.com/paper/2412.09235/) (2024)
- [DDEQs: Distributional Deep Equilibrium Models through Wasserstein Gradient Flows](https://mlsanity.com/paper/2503.01140/) (2025)
- [New universal operator approximation theorem for encoder-decoder architectures](https://mlsanity.com/paper/2503.24092/) (2025)
- [Schr"odinger bridge for generative AI: Soft-constrained formulation and convergence analysis](https://mlsanity.com/paper/2510.11829/) (2025)
- [Stability of the Kim--Milman flow map](https://mlsanity.com/paper/2511.01154/) (2025)
- [Hessian stability and convergence rates for entropic and Sinkhorn potentials via semiconcavity](https://mlsanity.com/paper/2504.11133/) (2025)
- [Near-Lipschitz stability of the Kim--Milman flow map](https://mlsanity.com/paper/2606.23383/) (2026)
- [Uniform Statistical Convergence of Empirical Sinkhorn Potentials with Exponential and Polynomial Dependence on the Regularization Parameter](https://mlsanity.com/paper/2608.29152/) (2026)
- [Low-Budget Active Learning through Entropic Optimal Transport](https://mlsanity.com/paper/2610.01199/) (2026)

## Abstract (arXiv)

The function that maps a family of probability measures to the solution of the dual entropic optimal transport problem is known as the Schr"odinger map. We prove that when the cost function is Ck+1 with k∈ mathbbN^* then this map is Lipschitz continuous from the L2-Wasserstein space to the space of Ck functions. Our result holds on compact domains and covers the multi-marginal case. We also include regularity results under negative Sobolev metrics weaker than Wasserstein under stronger smoothness assumptions on the cost. As applications, we prove displacement smoothness of the entropic optimal transport cost and the well-posedness of certain Wasserstein gradient flows involving this functional, including the Sinkhorn divergence and a multi-species system.