Least squares approximations in linear statistical inverse learning problems

TL;DR

This paper develops a theoretical framework for statistical inverse learning, demonstrating how to achieve optimal convergence rates for function reconstruction using finite-dimensional projections and maximum likelihood estimation.

Contribution

It introduces a novel analysis combining inverse learning with regularization via projections, establishing minimax optimal convergence rates for the ML estimator.

Findings

Probabilistic convergence rates for the ML estimator are derived.

Convergence rates in expectation are established with a norm-based cutoff.

The rates are proven to be minimax optimal.

Abstract

Statistical inverse learning aims at recovering an unknown function $f$ from randomly scattered and possibly noisy point evaluations of another function $g$, connected to $f$ via an ill-posed mathematical model. In this paper we blend statistical inverse learning theory with the classical regularization strategy of applying finite-dimensional projections. Our key finding is that coupling the number of random point evaluations with the choice of projection dimension, one can derive probabilistic convergence rates for the reconstruction error of the maximum likelihood (ML) estimator. Convergence rates in expectation are derived with a ML estimator complemented with a norm-based cut-off operation. Moreover, we prove that the obtained rates are minimax optimal.