  In this work we consider a generalized bilevel optimization framework for
solving inverse problems. We introduce fractional Laplacian as a regularizer to
improve the reconstruction quality, and compare it with the total variation
regularization. We emphasize that the key advantage of using fractional
Laplacian as a regularizer is that it leads to a linear operator, as opposed to
the total variation regularization which results in a nonlinear degenerate
operator. Inspired by residual neural networks, to learn the optimal strength
of regularization and the exponent of fractional Laplacian, we develop a
dedicated bilevel optimization neural network with a variable depth for a
general regularized inverse problem. We also draw some parallels between an
activation function in a neural network and regularization. We illustrate how
to incorporate various regularizer choices into our proposed network. As an
example, we consider tomographic reconstruction as a model problem and show an
improvement in reconstruction quality, especially for limited data, via
fractional Laplacian regularization. We successfully learn the regularization
strength and the fractional exponent via our proposed bilevel optimization
neural network. We observe that the fractional Laplacian regularization
outperforms total variation regularization. This is specially encouraging, and
important, in the case of limited and noisy data.
