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arXiv · 2408.10690

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

Andrea Aspri·Leon Frischauf·Otmar Scherzer
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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.

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Andrea Aspri, Leon Frischauf, Otmar Scherzer. 2024-08-20. Spectral Function Space Learning and Numerical Linear Algebra Networks for Solving Linear Inverse Problems. https://arxiv.org/abs/2408.10690

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