Spectral Function Space Learning and Numerical Linear Algebra Networks for Solving Linear Inverse Problems

TL;DR

This paper introduces a neural network-based spectral decomposition method for solving ill-conditioned linear inverse problems, combining classical linear algebra techniques with deep learning to improve stability and reconstruction accuracy.

Contribution

It presents a novel spectral function space learning approach that unifies Gram-Schmidt and PCA within neural networks for operator reconstruction.

Findings

Effective spectral decomposition of operators from training data

Neural network implementation of classical linear algebra procedures

Successful numerical reconstructions without physical model knowledge

Abstract

We consider solving a probably ill-conditioned linear operator equation, where the operator is not modeled by physical laws but is specified via training pairs (consisting of images and data) of the input-output relation of the operator. We derive a stable method for computing the operator, which consists of first a Gram-Schmidt orthonormalization of images and a principal component analysis of the data. This two-step algorithm provides a spectral decomposition of the linear operator. Moreover, we show that both Gram-Schmidt and principal component analysis can be written as a deep neural network, which relates this procedure to de-and encoder networks. Therefore, we call the two-step algorithm a linear algebra network. Finally, we provide numerical simulations showing the strategy is feasible for reconstructing spectral functions and for solving operator equations without explicitly exploiting the physical model.