{
  "id": "2402.04613",
  "version": "v1",
  "published": "2024-02-07T06:30:39.000Z",
  "updated": "2024-02-07T06:30:39.000Z",
  "title": "Wasserstein Gradient Flows for Moreau Envelopes of f-Divergences in Reproducing Kernel Hilbert Spaces",
  "authors": [
    "Sebastian Neumayer",
    "Viktor Stein",
    "Gabriele Steidl"
  ],
  "comment": "42 pages, 13 figures",
  "categories": [
    "stat.ML",
    "cs.LG",
    "math.FA",
    "math.OC"
  ],
  "abstract": "Most commonly used $f$-divergences of measures, e.g., the Kullback-Leibler divergence, are subject to limitations regarding the support of the involved measures. A remedy consists of regularizing the $f$-divergence by a squared maximum mean discrepancy (MMD) associated with a characteristic kernel $K$. In this paper, we use the so-called kernel mean embedding to show that the corresponding regularization can be rewritten as the Moreau envelope of some function in the reproducing kernel Hilbert space associated with $K$. Then, we exploit well-known results on Moreau envelopes in Hilbert spaces to prove properties of the MMD-regularized $f$-divergences and, in particular, their gradients. Subsequently, we use our findings to analyze Wasserstein gradient flows of MMD-regularized $f$-divergences. Finally, we consider Wasserstein gradient flows starting from empirical measures and provide proof-of-the-concept numerical examples with Tsallis-$\\alpha$ divergences.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-02-07T06:30:39.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "46N10",
      "46E22",
      "94A15"
    ],
    "keywords": [
      "reproducing kernel hilbert space",
      "moreau envelope",
      "analyze wasserstein gradient flows",
      "f-divergences",
      "exploit well-known results"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 42,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}