# The Riemannian geometry of Sinkhorn divergences

arXiv: https://arxiv.org/abs/2405.04987 · HTML: https://mlsanity.com/paper/2405.04987/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 8 May 2024.

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

- [Sinkhorn Treatment Effects: A Causal Optimal Transport Measure](https://mlsanity.com/paper/2605.08485/) (2026)
- [Langevin Diffusion Approximation to Same Marginal Schr"odinger Bridge](https://mlsanity.com/paper/2505.07647/) (2025)
- [GeoTTER: Leveraging Local Geometry of Optimal Transport for Zero-Shot Classification](https://mlsanity.com/paper/2609.13518/) (2026)
- [Low-Budget Active Learning through Entropic Optimal Transport](https://mlsanity.com/paper/2610.01199/) (2026)

## Abstract (arXiv)

We propose a new metric between probability measures on a compact metric space that mirrors the Riemannian manifold-like structure of quadratic optimal transport but includes entropic regularization. Its metric tensor is given by the Hessian of the Sinkhorn divergence, a debiased variant of entropic optimal transport. We precisely identify the tangent space it induces, which turns out to be related to a Reproducing Kernel Hilbert Space (RKHS). As usual in Riemannian geometry, the distance is built by looking for shortest paths. We prove that our distance is geodesic, metrizes the weak-star topology, and is equivalent to a RKHS norm. Still it retains the geometric flavor of optimal transport: as a paradigmatic example, translations are geodesics for the quadratic cost on ℝd. We also show two negative results on the Sinkhorn divergence that may be of independent interest: that it is not jointly convex, and that its square root is not a distance because it fails to satisfy the triangle inequality.