# Convergence of Gaussian-smoothed optimal transport distance with sub-gamma distributions and dependent samples

arXiv: https://arxiv.org/abs/2103.00394 · HTML: https://mlsanity.com/paper/2103.00394/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- Builds on 17 of its 55 references that are in the map.
- First posted on arXiv 28 Feb 2021.

## Builds on (most cited first, 17 of 17)

- [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)
- [Optimal Transport for Domain Adaptation](https://mlsanity.com/paper/1507.00504/) (2015)
- [Kernel Mean Embedding of Distributions: A Review and Beyond](https://mlsanity.com/paper/1605.09522/) (2016)
- [Generative Moment Matching Networks](https://mlsanity.com/paper/1502.02761/) (2015)
- [MMD GAN: Towards Deeper Understanding of Moment Matching Network](https://mlsanity.com/paper/1705.08584/) (2017)
- [DeepJDOT: Deep Joint Distribution Optimal Transport for Unsupervised Domain Adaptation](https://mlsanity.com/paper/1803.10081/) (2018)
- [Statistical bounds for entropic optimal transport: sample complexity and the central limit theorem](https://mlsanity.com/paper/1905.11882/) (2019)
- [Interpretable Distribution Features with Maximum Testing Power](https://mlsanity.com/paper/1605.06796/) (2016)
- [Differential Properties of Sinkhorn Approximation for Learning with Wasserstein Distance](https://mlsanity.com/paper/1805.11897/) (2018)
- [Fast Two-Sample Testing with Analytic Representations of Probability Measures](https://mlsanity.com/paper/1506.04725/) (2015)
- [Convergence and Concentration of Empirical Measures under Wasserstein Distance in Unbounded Functional Spaces](https://mlsanity.com/paper/1804.10556/) (2018)
- [Multiscale Strategies for Computing Optimal Transport](https://mlsanity.com/paper/1708.02469/) (2017)
- [FastMMD: Ensemble of Circular Discrepancy for Efficient Two-Sample Test](https://mlsanity.com/paper/1405.2664/) (2014)
- [Better and Simpler Error Analysis of the Sinkhorn-Knopp Algorithm for Matrix Scaling](https://mlsanity.com/paper/1801.02790/) (2018)
- [Greedy stochastic algorithms for entropy-regularized optimal transport problems](https://mlsanity.com/paper/1803.01347/) (2018)
- [Vector and Matrix Optimal Mass Transport: Theory, Algorithm, and Applications](https://mlsanity.com/paper/1712.10279/) (2017)

## Built on by (most cited first, 8 of 8)

- [Controlling Wasserstein Distances by Kernel Norms with Application to Compressive Statistical Learning](https://mlsanity.com/paper/2112.00423/) (2021)
- [Can Modifying Data Address Graph Domain Adaptation?](https://mlsanity.com/paper/2407.19311/) (2024)
- [Optimal Regularization for a Data Source](https://mlsanity.com/paper/2212.13597/) (2022)
- [Characterizing Dependence of Samples along the Langevin Dynamics and Algorithms via Contraction of Phi-Mutual Information](https://mlsanity.com/paper/2402.17067/) (2024)
- [Order Constraints in Optimal Transport](https://mlsanity.com/paper/2110.07275/) (2021)
- [Differentially Private Wasserstein Barycenters](https://mlsanity.com/paper/2510.03021/) (2025)
- [Limit Distribution Theory for the Smooth 1-Wasserstein Distance with Applications](https://mlsanity.com/paper/2107.13494/) (2021)
- [Nonparametric mixture MLEs under Gaussian-smoothed optimal transport distance](https://mlsanity.com/paper/2112.02421/) (2021)

## Abstract (arXiv)

The Gaussian-smoothed optimal transport (GOT) framework, recently proposed by Goldfeld et al., scales to high dimensions in estimation and provides an alternative to entropy regularization. This paper provides convergence guarantees for estimating the GOT distance under more general settings. For the Gaussian-smoothed p-Wasserstein distance in d dimensions, our results require only the existence of a moment greater than d + 2p. For the special case of sub-gamma distributions, we quantify the dependence on the dimension d and establish a phase transition with respect to the scale parameter. We also prove convergence for dependent samples, only requiring a condition on the pairwise dependence of the samples measured by the covariance of the feature map of a kernel space. A key step in our analysis is to show that the GOT distance is dominated by a family of kernel maximum mean discrepancy (MMD) distances with a kernel that depends on the cost function as well as the amount of Gaussian smoothing. This insight provides further interpretability for the GOT framework and also introduces a class of kernel MMD distances with desirable properties. The theoretical results are supported by numerical experiments.