Work

Least Squares Approximations in Linear Statistical Inverse Learning Problems

Year: 2024

Type: article

Abstract: . Statistical inverse learning aims at recovering an unknown function [Formula: see text] from randomly scattered and possibly noisy point evaluations of another function [Formula: see text], connected to [Formula: see text] 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.

Source: SIAM Journal on Numerical Analysis

DOI: 10.1137/22m1538600

Author: Tapio Helin

Institution: Lappeenranta-Lahti University of Technology

Language: English

Topic: Numerical methods in inverse problems

Subfield: Mathematical Physics

Field: Mathematics

Domain: Physical Sciences

Open Access status: green

Funder: Academy of Finland

Award: Uncertainty quantification in non-linear inverse problems