Convergence rate of general entropic optimal transport costs
A talk in the Oberseminar Analysis series by
Luca Tamanini
| Abstract: | Entropic Optimal Transport consists in the minimization of a transport functional penalized by an entropy term. When the penalization/noise parameter vanishes, the original transport problem is recovered and this remark yields several natural questions, as for instance the convergence rate of the optimal value. This has recently been the subject of a deep investigation and the existing literature covers the quadratic case up to the second order [Conforti-T. '21] and suitable smooth perturbations of the quadratic cost up to the first order [Pal '19]. Aim of this talk is to deal with more general cost functions. In particular, certain costs for which the solutions of the unregularized problem are not induced by a transport map and are not even unique are covered. After reviewing the existing literature, on the one hand, under very mild assumptions we will obtain a sharp upper bound on the gap between transport and EOT costs in terms of the so-called information or Rényi dimension. On the other hand, under an infinitesimal twist condition on the cost and relying on Minty's trick we will show a quadratic detachment of the duality gap, which yields the matching lower bound. (based on a joint work with G. Carlier and P. Pegon) Within the CRC this talk is associated to the project(s): A8 |