HDI Lab: On Optimal Matching of Random Samples

Speaker: Michel Ledoux (Université de Toulouse)

Abstract: Optimal matching problems are random variational problems widely investigated in the mathematics and physics literature, and in probability theory and statistics in the study of rates of convergence of empirical measures. Two-dimensional matching of uniform samples gave in particular rise to deep results investigated from various view points (combinatorial, generic chaining, Fourier analysis). After a review of known results, we investigate more precisely the case of Gaussian samples, first in dimension one, on the basis of explicit representations of Kantorovich metrics and a sharp analysis of more general log-concave distributions in terms of their isoperimetric profile (joint work with S. Bobkov), and then in dimension two and higher following the PDE and transportation approach recently put forward by L. Ambrosio, F. Stra and D. Trevisan.