mlsanityData to 7 Oct 2026

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

arXiv 2112.00423 ↗First posted 1 Dec 2021stat.ML cs.LGcited by 13

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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.

Key facts. Cited by 13 papers in the map (data through 7 Oct 2026). Builds on 27 of its 85 references that are in the map. First posted on arXiv 1 Dec 2021.

Builds on

27 of 85 references.

References · 27

  1. 2018Computational Optimal Transportcited by 765
  2. 2017Towards Principled Methods for Training Generative Adversarial Networkscited by 460
  3. 2018Lipschitz regularity of deep neural networks: analysis and efficient estimationcited by 413
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15 more
  1. 2019Statistical Inference for Generative Models with Maximum Mean Discrepancycited by 43
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  5. 2021Smooth p-Wasserstein Distance: Structure, Empirical Approximation, and Statistical Applicationscited by 15
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Citing papers per year: 2022: 1; 2023: 2; 2024: 4; 2025: 3; 2026: 3.

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  1. 2023Generative Sliced MMD Flows with Riesz Kernelscited by 36
  2. 2024Wasserstein Gradient Flows for Moreau Envelopes of f-Divergences in Reproducing Kernel Hilbert Spacescited by 15
  3. 2024Generative neural networks for characteristic functionscited by 3
  4. 2023PACER: A Fully Push-forward-based Distributional Reinforcement Learning Algorithmcited by 2
  5. 2022Compressive Clustering with an Optical Processing Unitcited by 1
  6. 2024LaCoOT: Layer Collapse through Optimal Transportcited by 1
  7. 2025Scalable Learning from Probability Measures with Mean Measure Quantizationcited by 1
  8. 2025Quantitative convergence of trained single layer neural networks to Gaussian processescited by 1
  9. 2024Utilizing Maximum Mean Discrepancy Barycenter for Propagating the Uncertainty of Value Functions in Reinforcement Learningcited by 0
  10. 2025MMD-Newton Method for Multi-objective Optimizationcited by 0
  11. 2026Efficient Techniques for Data Reconstruction, with Finite-Width Recovery Guaranteescited by 0
  12. 2026ParetoTransport: Generative Optimization by Mass Transport Toward The Pareto Frontcited by 0
1 more
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