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

## TL;DR

This paper introduces a bilevel optimization neural network that learns optimal fractional Laplacian regularization parameters for inverse problems, demonstrating improved tomographic reconstruction especially with limited data.

## Contribution

It proposes a novel bilevel neural network framework that learns regularization strength and fractional Laplacian exponent, outperforming total variation in inverse problems.

## Key findings

- Fractional Laplacian regularization improves reconstruction quality.
- The neural network effectively learns optimal regularization parameters.
- Performance surpasses total variation regularization, especially with limited data.

## Abstract

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.