2024/08/20 by Andrea Aspri, Aspri, Andrea, Leon Frischauf +3 · 1 citation
Computer Science · Mathematics · #FOS: Mathematics #Functional Analysis (math.FA) #Neural Networks and Applications #Numerical Analysis (math.NA) #Numerical methods in inverse problems #Statistical and numerical algorithms
paper · pdf · doi:10.48550/arxiv.2408.10690
openalex publication_date 2024/08/20 · openalex created_date 2024/10/01 · openalex updated_date 2026/07/28
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.