Speaker
Sven Wang, Humboldt University Berlin
Abstract
We present statistical convergence results for the statistical learning of non-linear operators between infinite-dimensional spaces. Given a possibly nonlinear map between two separable Hilbert spaces, we analyze the problem of recovering the map from noisy input-output pairscorrupted by i.i.d. white noise processes or subgaussian random variables. We provide a general convergence results for least-squares-type empirical risk minimizers over compact regression classes, in terms of their approximation properties and metric entropy bounds, proved using empirical process theory. This extends classical results in finite-dimensional nonparametric regression to an infinite-dimensional setting. As a concrete application, we study an encoder-decoder based neural operator architecture. Assuming holomorphy of the operator, we prove algebraic (in the sample size) convergence rates in this setting, thereby overcoming the curse of dimensionality. To illustrate the wide applicability of our results, we discuss a parametric Darcy-flow problem on the torus.