# Controlling Wasserstein Distances by Kernel Norms with Application to Compressive Statistical Learning

arXiv: https://arxiv.org/abs/2112.00423 · HTML: https://mlsanity.com/paper/2112.00423/

## Key facts

- Cited by 14 papers in the map (data through 14 Oct 2026).
- Builds on 27 of its 85 references that are in the map.
- First posted on arXiv 1 Dec 2021.

## Builds on (most cited first, 27 of 27)

- [Computational Optimal Transport](https://mlsanity.com/paper/1803.00567/) (2018)
- [Towards Principled Methods for Training Generative Adversarial Networks](https://mlsanity.com/paper/1701.04862/) (2017)
- [Lipschitz regularity of deep neural networks: analysis and efficient estimation](https://mlsanity.com/paper/1805.10965/) (2018)
- [Kernel Mean Embedding of Distributions: A Review and Beyond](https://mlsanity.com/paper/1605.09522/) (2016)
- [Learning Generative Models with Sinkhorn Divergences](https://mlsanity.com/paper/1706.00292/) (2017)
- [Efficient and Accurate Estimation of Lipschitz Constants for Deep Neural Networks](https://mlsanity.com/paper/1906.04893/) (2019)
- [Learning with a Wasserstein Loss](https://mlsanity.com/paper/1506.05439/) (2015)
- [The Lipschitz Constant of Self-Attention](https://mlsanity.com/paper/2006.04710/) (2020)
- [Random Features for Kernel Approximation: A Survey on Algorithms, Theory, and Beyond](https://mlsanity.com/paper/2004.11154/) (2020)
- [Lipschitz constant estimation of Neural Networks via sparse polynomial optimization](https://mlsanity.com/paper/2004.08688/) (2020)
- [On the Error of Random Fourier Features](https://mlsanity.com/paper/1506.02785/) (2015)
- [Generalized Low Rank Models](https://mlsanity.com/paper/1410.0342/) (2014)
- [Statistical Inference for Generative Models with Maximum Mean Discrepancy](https://mlsanity.com/paper/1906.05944/) (2019)
- [Optimal Rates for Random Fourier Features](https://mlsanity.com/paper/1506.02155/) (2015)
- [Metrizing Weak Convergence with Maximum Mean Discrepancies](https://mlsanity.com/paper/2006.09268/) (2020)
- [On gradient regularizers for MMD GANs](https://mlsanity.com/paper/1805.11565/) (2018)
- [Smooth p-Wasserstein Distance: Structure, Empirical Approximation, and Statistical Applications](https://mlsanity.com/paper/2101.04039/) (2021)
- [Sketching for Large-Scale Learning of Mixture Models](https://mlsanity.com/paper/1606.02838/) (2016)
- [Differentially Private Database Release via Kernel Mean Embeddings](https://mlsanity.com/paper/1710.01641/) (2017)
- [Compressive Statistical Learning with Random Feature Moments](https://mlsanity.com/paper/1706.07180/) (2017)
- [Compressive K-means](https://mlsanity.com/paper/1610.08738/) (2016)
- [Statistical Learning Guarantees for Compressive Clustering and Compressive Mixture Modeling](https://mlsanity.com/paper/2004.08085/) (2020)
- [Convergence of Gaussian-smoothed optimal transport distance with sub-gamma distributions and dependent samples](https://mlsanity.com/paper/2103.00394/) (2021)
- [Mean Nystr"om Embeddings for Adaptive Compressive Learning](https://mlsanity.com/paper/2110.10996/) (2021)
- [Instance Optimal Decoding and the Restricted Isometry Property](https://mlsanity.com/paper/1802.09905/) (2018)
- [Compressive Classification (Machine Learning without learning)](https://mlsanity.com/paper/1812.01410/) (2018)
- [Compressive Learning of Generative Networks](https://mlsanity.com/paper/2002.05095/) (2020)

## Built on by (most cited first, 14 of 14)

- [Generative Sliced MMD Flows with Riesz Kernels](https://mlsanity.com/paper/2305.11463/) (2023)
- [Wasserstein Gradient Flows for Moreau Envelopes of f-Divergences in Reproducing Kernel Hilbert Spaces](https://mlsanity.com/paper/2402.04613/) (2024)
- [Generative neural networks for characteristic functions](https://mlsanity.com/paper/2401.04778/) (2024)
- [PACER: A Fully Push-forward-based Distributional Reinforcement Learning Algorithm](https://mlsanity.com/paper/2306.06637/) (2023)
- [Compressive Clustering with an Optical Processing Unit](https://mlsanity.com/paper/2206.05928/) (2022)
- [LaCoOT: Layer Collapse through Optimal Transport](https://mlsanity.com/paper/2406.08933/) (2024)
- [Scalable Learning from Probability Measures with Mean Measure Quantization](https://mlsanity.com/paper/2502.04907/) (2025)
- [Quantitative convergence of trained single layer neural networks to Gaussian processes](https://mlsanity.com/paper/2509.24544/) (2025)
- [Utilizing Maximum Mean Discrepancy Barycenter for Propagating the Uncertainty of Value Functions in Reinforcement Learning](https://mlsanity.com/paper/2404.00686/) (2024)
- [MMD-Newton Method for Multi-objective Optimization](https://mlsanity.com/paper/2505.14610/) (2025)
- [Efficient Techniques for Data Reconstruction, with Finite-Width Recovery Guarantees](https://mlsanity.com/paper/2605.06519/) (2026)
- [ParetoTransport: Generative Optimization by Mass Transport Toward The Pareto Front](https://mlsanity.com/paper/2609.07706/) (2026)
- [Generative Priors Conditioned on Natural Language for Bayesian Inversion in PDEs](https://mlsanity.com/paper/2609.32941/) (2026)
- [Markov Chain Convergence in Kernel-Induced Geometry](https://mlsanity.com/paper/2610.13916/) (2026)

## Abstract (arXiv)

Comparing probability distributions is at the crux of many machine learning algorithms. Maximum Mean Discrepancies (MMD) and Wasserstein distances are two classes of distances between probability distributions that have attracted abundant attention in past years. This paper establishes some conditions under which the Wasserstein distance can be controlled by MMD norms. Our work is motivated by the compressive statistical learning (CSL) theory, a general framework for resource-efficient large scale learning in which the training data is summarized in a single vector (called sketch) that captures the information relevant to the considered learning task. Inspired by existing results in CSL, we introduce the H"older Lower Restricted Isometric Property and show that this property comes with interesting guarantees for compressive statistical learning. Based on the relations between the MMD and the Wasserstein distances, we provide guarantees for compressive statistical learning by introducing and studying the concept of Wasserstein regularity of the learning task, that is when some task-specific metric between probability distributions can be bounded by a Wasserstein distance.