{
  "id": "1901.04568",
  "version": "v1",
  "published": "2019-01-14T21:19:17.000Z",
  "updated": "2019-01-14T21:19:17.000Z",
  "title": "Kantorovich vs. Monge: A Numerical Classification of Extremal Multi-Marginal Mass Transports on Finite State Spaces",
  "authors": [
    "Daniela Vögler"
  ],
  "categories": [
    "math.AP",
    "math.OC"
  ],
  "abstract": "We analyze the validity of Monge's ansatz regarding the symmetric multi-marginal Kantorovich optimal transport problem on finite state spaces, with uniform-marginal constraint. The class of Monge states reduces the number of unknowns from combinatorial in both $N$ and $\\ell$ to linear in $N$ and $\\ell$, where $N$ is the number of marginals and $\\ell$ the number of states. Unfortunately, this low-dimensional ansatz space is insufficient, i.e., there exist cost functions such that the corresponding multi-marginal optimal transport problem does not admit a Monge-type optimizer. We will analyze this insufficiency numerically by utilizing the convex geometry of the set of admissible trial states for symmetric respectively pairwise-symmetric cost functions. We further consider a model problem of optimally coupling $N$ marginals on three states with respect to a pairwise-symmetric cost function. The restriction to a state space of three elements allows us to visually compare Kantorovich's to Monge's ansatz space. This visual comparison includes a consideration of the volumetric ratio.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-01-14T21:19:17.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "49J40",
      "49K30",
      "52B12",
      "49S05"
    ],
    "keywords": [
      "extremal multi-marginal mass transports",
      "finite state spaces",
      "multi-marginal optimal transport problem",
      "kantorovich optimal transport problem",
      "multi-marginal kantorovich optimal transport"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}