Sieve Estimation of Optimal Transport Maps from Paired Data in Gaussian Spaces

Abstract

We estimate optimal transport maps on an infinite-dimensional Hilbert space with a Gaussian reference measure, from noisy paired observations. A source draw is seen together with a noisy evaluation of its image, rather than through independent unpaired samples. The estimator is a cylindrical sieve of Cameron--Martin gradient maps, restricted to a compact parameter set; individual sieve elements need not be transport maps. It yields a finite regression contrast even though the noise has infinite Cameron--Martin norm, and reduces estimation to finite-dimensional empirical risk minimization. We establish a nonasymptotic oracle inequality separating approximation error, stochastic error and the local conditioning of the parametrization, together with a minimax lower bound of order $N^{-s/(2s+1)}$ under weighted coordinate regularity of order $s$. Output regularity alone does not deliver cylindrical approximation; for general Sobolev potentials an input-regularity index does, via a conditional Gaussian Poincar\'e argument, and for potentials of bounded chaos degree the degree bound plays that role, an orthogonal Hermite sieve then attaining the same rate with the interaction order replacing unity in the exponent. That rate is minimax on a diagonal Gaussian class and on a nonlinear block class whose interaction survives every fixed orthogonal change of coordinates.