Learning linear operators: Infinite-dimensional regression as a well-behaved non-compact inverse problem

TL;DR

This paper studies the problem of learning linear operators between Hilbert spaces from data, reformulating it as a non-compact inverse problem and deriving dimension-free learning rates under certain conditions.

Contribution

It introduces a novel inverse problem perspective for infinite-dimensional regression, establishing spectral equivalences and deriving new learning rates for operator estimation.

Findings

Inverse problem for operator learning is spectrally equivalent to scalar regression inverse problems.

Dimension-free rates are derived for generic algorithms under Hölder source conditions.

Rates match classical kernel regression results for scalar responses.

Abstract

We consider the problem of learning a linear operator $\theta$ between two Hilbert spaces from empirical observations, which we interpret as least squares regression in infinite dimensions. We show that this goal can be reformulated as an inverse problem for $\theta$ with the feature that its forward operator is generally non-compact (even if $\theta$ is assumed to be compact or of $p$-Schatten class). However, we prove that, in terms of spectral properties and regularisation theory, this inverse problem is equivalent to the known compact inverse problem associated with scalar response regression. Our framework allows for the elegant derivation of dimension-free rates for generic learning algorithms under H\"older-type source conditions. The proofs rely on the combination of techniques from kernel regression with recent results on concentration of measure for sub-exponential Hilbertian random variables. The obtained rates hold for a variety of practically-relevant scenarios in functional regression as well as nonlinear regression with operator-valued kernels and match those of classical kernel regression with scalar response.