# Finite sample approximations of exact and entropic Wasserstein distances between covariance operators and Gaussian processes

arXiv: https://arxiv.org/abs/2104.12368 · HTML: https://mlsanity.com/paper/2104.12368/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 9 of its 52 references that are in the map.
- First posted on arXiv 26 Apr 2021.

## Builds on (most cited first, 9 of 9)

- [Learning Generative Models with Sinkhorn Divergences](https://mlsanity.com/paper/1706.00292/) (2017)
- [Functional Variational Bayesian Neural Networks](https://mlsanity.com/paper/1903.05779/) (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)
- [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)
- [Convergence and finite sample approximations of entropic regularized Wasserstein distances in Gaussian and RKHS settings](https://mlsanity.com/paper/2101.01429/) (2021)
- [Sinkhorn AutoEncoders](https://mlsanity.com/paper/1810.01118/) (2018)
- [Estimating 2-Sinkhorn Divergence between Gaussian Processes from Finite-Dimensional Marginals](https://mlsanity.com/paper/2102.03267/) (2021)

## Abstract (arXiv)

This work studies finite sample approximations of the exact and entropic regularized Wasserstein distances between centered Gaussian processes and, more generally, covariance operators of functional random processes. We first show that these distances/divergences are fully represented by reproducing kernel Hilbert space (RKHS) covariance and cross-covariance operators associated with the corresponding covariance functions. Using this representation, we show that the Sinkhorn divergence between two centered Gaussian processes can be consistently and efficiently estimated from the divergence between their corresponding normalized finite-dimensional covariance matrices, or alternatively, their sample covariance operators. Consequently, this leads to a consistent and efficient algorithm for estimating the Sinkhorn divergence from finite samples generated by the two processes. For a fixed regularization parameter, the convergence rates are it dimension-independent and of the same order as those for the Hilbert-Schmidt distance. If at least one of the RKHS is finite-dimensional, we obtain a it dimension-dependent sample complexity for the exact Wasserstein distance between the Gaussian processes.