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In this work we consider a generalized bilevel optimization framework forsolving inverse problems. We introduce fractional Laplacian as a regularizer toimprove the reconstruction quality, and compare it with the total variationregularization. We emphasize that the key advantage of using fractionalLaplacian as a regularizer is that it leads to a linear operator, as opposed tothe total variation regularization which results in a nonlinear degenerateoperator. Inspired by residual neural networks, to learn the optimal strengthof regularization and the exponent of fractional Laplacian, we develop adedicated bilevel optimization neural network with a variable depth for ageneral regularized inverse problem. We also draw some parallels between anactivation function in a neural network and regularization. We illustrate howto incorporate various regularizer choices into our proposed network. As anexample, we consider tomographic reconstruction as a model problem and show animprovement in reconstruction quality, especially for limited data, viafractional Laplacian regularization. We successfully learn the regularizationstrength and the fractional exponent via our proposed bilevel optimizationneural network. We observe that the fractional Laplacian regularizationoutperforms total variation regularization. This is specially encouraging, andimportant, in the case of limited and noisy data.

Bilevel Optimization, Deep Learning and Fractional Laplacian Regularization with Applications in Tomography