{
  "id": "2406.10823",
  "version": "v1",
  "published": "2024-06-16T07:23:26.000Z",
  "updated": "2024-06-16T07:23:26.000Z",
  "title": "Iterated Schrödinger bridge approximation to Wasserstein Gradient Flows",
  "authors": [
    "Medha Agarwal",
    "Zaid Harchaoui",
    "Garrett Mulcahy",
    "Soumik Pal"
  ],
  "comment": "36 pages, 1 figure",
  "categories": [
    "math.PR",
    "stat.ML"
  ],
  "abstract": "We introduce a novel discretization scheme for Wasserstein gradient flows that involves successively computing Schr\\\"{o}dinger bridges with the same marginals. This is different from both the forward/geodesic approximation and the backward/Jordan-Kinderlehrer-Otto (JKO) approximations. The proposed scheme has two advantages: one, it avoids the use of the score function, and, two, it is amenable to particle-based approximations using the Sinkhorn algorithm. Our proof hinges upon showing that relative entropy between the Schr\\\"{o}dinger bridge with the same marginals at temperature $\\epsilon$ and the joint distribution of a stationary Langevin diffusion at times zero and $\\epsilon$ is of the order $o(\\epsilon^2)$ with an explicit dependence given by Fisher information. Owing to this inequality, we can show, using a triangular approximation argument, that the interpolated iterated application of the Schr\\\"{o}dinger bridge approximation converge to the Wasserstein gradient flow, for a class of gradient flows, including the heat flow. The results also provide a probabilistic and rigorous framework for the convergence of the self-attention mechanisms in transformer networks to the solutions of heat flows, first observed in the inspiring work SABP22 in machine learning research.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-06-16T07:23:26.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "49N99",
      "49Q22",
      "60J60"
    ],
    "keywords": [
      "wasserstein gradient flow",
      "iterated schrödinger bridge approximation",
      "heat flow",
      "novel discretization scheme",
      "bridge approximation converge"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 36,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}