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Comparing probability distributions is at the crux of many machine learningalgorithms. Maximum Mean Discrepancies (MMD) and Wasserstein distances are twoclasses of distances between probability distributions that have attractedabundant attention in past years. This paper establishes some conditions underwhich the Wasserstein distance can be controlled by MMD norms. Our work ismotivated by the compressive statistical learning (CSL) theory, a generalframework for resource-efficient large scale learning in which the trainingdata is summarized in a single vector (called sketch) that captures theinformation relevant to the considered learning task. Inspired by existingresults in CSL, we introduce the H\"older Lower Restricted Isometric Propertyand show that this property comes with interesting guarantees for compressivestatistical learning. Based on the relations between the MMD and theWasserstein distances, we provide guarantees for compressive statisticallearning by introducing and studying the concept of Wasserstein regularity ofthe learning task, that is when some task-specific metric between probabilitydistributions can be bounded by a Wasserstein distance.

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