Spectral Function Space Learning and Numerical Linear Algebra Networks for Solving Linear Inverse Problems
<|reference_start|>Spectral Function Space Learning and Numerical Linear Algebra Networks for Solving Linear Inverse Problems: 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.<|reference_end|>
@article{aspri2024spectral,
    title={Spectral Function Space Learning and Numerical Linear Algebra Networks
  for Solving Linear Inverse Problems},
    author={Andrea Aspri, Leon Frischauf, Otmar Scherzer},
    journal={arXiv preprint arXiv:2408.10690},
    year={2024},
    archivePrefix={arXiv},
    eprint={2408.10690},
    primaryClass={math.NA cs.NA math.FA}
}