# Stability of optimal transport maps on Riemannian manifolds

arXiv: https://arxiv.org/abs/2504.05412 · HTML: https://mlsanity.com/paper/2504.05412/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 7 Apr 2025.

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

- [Statistical Inference for Optimal Transport Maps: Recent Advances and Perspectives](https://mlsanity.com/paper/2506.19025/) (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)

## Abstract (arXiv)

We prove quantitative bounds on the stability of optimal transport maps and Kantorovich potentials from a fixed source measure ρ under variations of the target measure μ, when the cost function is the squared Riemannian distance on a Riemannian manifold. Previous works were restricted to subsets of Euclidean spaces, or made specific assumptions either on the manifold, or on the regularity of the transport maps. Our proof techniques combine entropy-regularized optimal transport with spectral and integral-geometric techniques. As some of the arguments do not rely on the Riemannian structure, our work also paves the way towards understanding stability of optimal transport in more general geometric spaces.