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

Abstract

We study the estimation of infinite-dimensional optimal transport maps from noisy paired observations. The population map pushes a Gaussian reference measure forward to a target probability measure on a function space and takes the Cameron--Martin gradient form $T=I+{\nabla }_{\mathcal{H}}\varphi$. Our estimator uses cylindrical gradient sieves based on finitely many Cameron--Martin coordinates, thereby reducing the problem to finite-dimensional empirical risk minimization. A local nonasymptotic oracle inequality separates cylindrical approximation and statistical estimation errors while accounting for the conditioning of the parametrization. The approximation analysis relies on regularity conditions governing coordinate decay and dependence on omitted input coordinates. For diagonal Gaussian and nonlinear block-interaction classes, we derive matching upper and lower bounds that establish the minimax rate $N^{-s/(2s+1)}$ in expected norm, where $s$ measures weighted coordinate regularity. We further analyze a continuous two-groups model with Gaussian--Laplace mixtures and derive a prediction-risk bound for the resulting transport-map estimator.