Lecture 1: An introduction to Monge–Kantorovich optimal transport and entropic relaxations
The mathematics of Monge–Kantorovich optimal transport (OT) has grown to be a unifying theme in many scientific disciplines, from purely mathematical areas such as analysis, geometry, and probability to revolutionary new methods in economics, statistics, machine learning and artificial intelligence. Much of these are due to impressive leaps in computational methods in OT that hinge on entropy based approximations. For dynamical OT such relaxations correspond to stochastic processes called Schrödinger bridges. This is an introductory talk on OT, entropic relaxations, and Schrödinger bridges that will attempt to give a flavor of this "hot area" by sewing through various recent advances.
Lecture 2: On the difference between entropic cost and the optimal transport cost
Consider a Monge–Kantorovich problem of transporting densities with a strictly convex cost function. The entropic relaxation cost is known to converge to the cost of optimal transport. We are interested in the difference between the two, suitably scaled. We show that this limit is always given by the relative entropy of the target density with respect to a Riemannian volume measure that measures the local sensitivity of the Monge map. In the special case of the quadratic Wasserstein transport, this relative entropy is exactly one half of the difference of entropies of the target and initial densities. The proofs are based of Gaussian approximations to Schrödinger bridges which can be interpreted as a higher order large deviation.