Gradient flows in the geometry of Sinkhorn divergences

What happens to Wasserstein gradient flows if one uses entropic optimal transport instead of classical optimal transport? I will explain why it may be relevant to use Sinkhorn divergences, built on entropic optimal transport, as they allow the regularization parameter to remain fixed. This leads to the study of the Riemannian geometry induced by the Sinkhorn divergences: it retains some features of optimal transport geometry while being “smoother.” The gradient flows of potential energies in this geometry exhibit some intriguing features, which I will detail. This is joint work with Mathis Hardion, Jonas Luckhardt, Gilles Mordant, Bernhard Schmitzer and Luca Tamanini.