# Convergence and finite sample approximations of entropic regularized Wasserstein distances in Gaussian and RKHS settings

arXiv: https://arxiv.org/abs/2101.01429 · HTML: https://mlsanity.com/paper/2101.01429/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 56 references that are in the map.
- First posted on arXiv 5 Jan 2021.

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

- [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)
- [A survey of the Schr"odinger problem and some of its connections with optimal transport](https://mlsanity.com/paper/1308.0215/) (2013)
- [Stochastic Optimization for Large-scale Optimal Transport](https://mlsanity.com/paper/1605.08527/) (2016)
- [On Wasserstein Two Sample Testing and Related Families of Nonparametric Tests](https://mlsanity.com/paper/1509.02237/) (2015)
- [Adversarial Regularizers in Inverse Problems](https://mlsanity.com/paper/1805.11572/) (2018)
- [Statistical bounds for entropic optimal transport: sample complexity and the central limit theorem](https://mlsanity.com/paper/1905.11882/) (2019)
- [Entropy-Regularized 2-Wasserstein Distance between Gaussian Measures](https://mlsanity.com/paper/2006.03416/) (2020)
- [Procrustes Metrics on Covariance Operators and Optimal Transportation of Gaussian Processes](https://mlsanity.com/paper/1801.01990/) (2018)
- [How Well Do WGANs Estimate the Wasserstein Metric?](https://mlsanity.com/paper/1910.03875/) (2019)
- [The statistical effect of entropic regularization in optimal transportation](https://mlsanity.com/paper/2006.05199/) (2020)
- [Entropic regularization of Wasserstein distance between infinite-dimensional Gaussian measures and Gaussian processes](https://mlsanity.com/paper/2011.07489/) (2020)

## Built on by (most cited first, 3 of 3)

- [Kullback-Leibler and Renyi divergences in reproducing kernel Hilbert space and Gaussian process settings](https://mlsanity.com/paper/2207.08406/) (2022)
- [Estimation of Riemannian distances between covariance operators and Gaussian processes](https://mlsanity.com/paper/2108.11683/) (2021)
- [Finite sample approximations of exact and entropic Wasserstein distances between covariance operators and Gaussian processes](https://mlsanity.com/paper/2104.12368/) (2021)

## Abstract (arXiv)

This work studies the convergence and finite sample approximations of entropic regularized Wasserstein distances in the Hilbert space setting. Our first main result is that for Gaussian measures on an infinite-dimensional Hilbert space, convergence in the 2-Sinkhorn divergence is it strictly weaker than convergence in the exact 2-Wasserstein distance. Specifically, a sequence of centered Gaussian measures converges in the 2-Sinkhorn divergence if the corresponding covariance operators converge in the Hilbert-Schmidt norm. This is in contrast to the previous known result that a sequence of centered Gaussian measures converges in the exact 2-Wasserstein distance if and only if the covariance operators converge in the trace class norm. In the reproducing kernel Hilbert space (RKHS) setting, the it kernel Gaussian-Sinkhorn divergence, which is the Sinkhorn divergence between Gaussian measures defined on an RKHS, defines a semi-metric on the set of Borel probability measures on a Polish space, given a characteristic kernel on that space. With the Hilbert-Schmidt norm convergence, we obtain it dimension-independent convergence rates for finite sample approximations of the kernel Gaussian-Sinkhorn divergence, with the same order as the Maximum Mean Discrepancy. These convergence rates apply in particular to Sinkhorn divergence between Gaussian measures on Euclidean and infinite-dimensional Hilbert spaces. The sample complexity for the 2-Wasserstein distance between Gaussian measures on Euclidean space, while dimension-dependent and larger than that of the Sinkhorn divergence, is exponentially faster than the worst case scenario in the literature.