B0785
Title: Convergence of a bivariate quantile transform, contours and local depth
Authors: Philippe Berthet - Toulouse University (France) [presenting]
Jean-Claude Fort - University Paris City CNRS (France)
Abstract: Given two samples of bivariate distributions P and Q, an empirical quantile transform is constructed that is easy to compute and converges to a transport map from P to Q. The keystone is a generator $G(P)$ based on the Kendall geometry of P, made of two families of mass curves indexed by $(0,1)$. Intersecting two $G(P)$ curves $(u1,u2)$ chosen uniformly yields a point X with distribution P. The coupling $[G(P), G(Q)](u)$ has the property to optimally transport the conditional laws of P and Q on their corresponding curves with respect to a large class of costs, including Wasserstein ones. The continuous transport built from the empirical counterpart of the generators can be computed in practice with samples of several million. Various notions of contours, global or local depth, follow since what is learned is the mass geometry. Gaussian process approximations are derived of a geometrical nature, involving non-independent global and local empirical processes along curves. This tool implies explicit CLTs for new non-parametric statistics. In particular, weak convergence of transport costs, contours, clusterings, local modes, trimmed supports and local depth fields is now accessible.
