Georgia Institute of Technology College of Sciences

Weighted Poincar\'e inequalities known for various laws such as the exponential or Cauchy ones are shown to follow from the "usual"  Poincar\'e inequality involving the non-local gradient.  A key ingredient in showing so is a covariance representation and Hardy's inequality.  

The framework under study is quite general and comprises infinitely divisible laws as well as some log-concave ones.  This same covariance representation is then used to obtain quantitative concentration inequalities of exponential type, recovering in particular the Gaussian results.  

Joint Work with Benjamin Arras.  

We present a unified methodology for obtaining rates of estimation of optimal transport maps in general function spaces. Our assumptions are significantly weaker than those appearing in the literature: we require only that the source measure P satisfy a Poincare inequality and that the optimal map be the gradient of a smooth convex function that lies in a space whose metric entropy can be controlled. As a special case, we recover known estimation rates for Holder transport maps, but also obtain nearly sharp results in many settings not covered by prior work. For example, we provide the first statistical rates of estimation when P is the normal distribution, between log-smooth and strongly log-concave distributions, and when the transport map is given by an infinite-width shallow neural network. (joint with Vincent Divol and Aram-Alexandre Pooladian.)