Monotone measure-transportation maps in Hilbert spaces, with statistical applications

Abstract

The contribution of this work is twofold. The first part deals with a Hilbert-space version of McCann’s celebrated result on the existence and uniqueness of monotone measure-preserving maps: given two probability measures P and Q on a separable Hilbert space H where P does not give mass to “small” sets (namely, Lipschitz hypersurfaces), we show, without imposing any moment assumptions, that there exists a gradient of convex function ∇ψ pushing P forward to Q. In case H is infinite-dimensional, P-a.e. uniqueness is not guaranteed, though; we show that uniqueness holds among all gradients of convex functions ∇ψ pushing P forward to Q for which the boundary of the domain of ψ has P-probability zero. This condition (hence the uniqueness of the gradient of convex function ∇ψ pushing P forward to Q) is automatically satisfied in the finite-dimensional case or when Q is boundedly supported (a natural assumption in several statistical applications). Furthermore, we establish stability results for transport maps in the sense of uniform convergence over compact “regularity sets.” As a consequence, we obtain a central limit theorem for the fluctuations of the optimal quadratic transport cost in a separable Hilbert space. In the second part of the paper we consider several important statistical applications of our results—center-outward ranks and quantiles for Hilbert-space-valued data, nonparametric distribution-free testing, and the construction of quantile regions. We show that the measure-transportation-based ranks are distribution-free and maximal ancillary, while the corresponding quantile functions fully characterize the underlying probability measures. These are the first notions of ranks and quantiles in non-locally compact spaces satisfying these properties.