Least Squares Approximations in Linear Statistical Inverse Learning Problems

SIAM Journal on Numerical Analysis, Volume 62, Issue 4, Page 2025-2047, August 2024.
Abstract. Statistical inverse learning aims at recovering an unknown function [math] from randomly scattered and possibly noisy point evaluations of another function [math], connected to [math] 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 cutoff operation. Moreover, we prove that the obtained rates are minimax optimal.