Spectral function space learning and numerical linear algebra networks for solving linear inverse problems

We consider solving an 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. The proposed method for computing the operator from training pairs consists of a Gram–Schmidt orthonormalization of images and a principal component analysis of data. Interestingly, this two-step algorithm provides us with a spectral decomposition of the linear operator, without explicit knowledge of it. Moreover, we indicate that both Gram–Schmidt and principal component analyses can be expressed as a deep neural network which delivers orthonormalized vectors from a set of vectors. This relates the algorithm to decoder and encoder networks. Therefore, we call the two-step algorithm a linear algebra network. Finally, we provide numerical simulations showing that the strategy is feasible for reconstructing spectral functions and for solving operator equations without explicitly making use of a physical model.