# Sparse synthesis regularization with deep neural networks

## TL;DR

This paper introduces a novel sparse reconstruction method using deep neural networks with an $ ext{l}^1$ penalty, enabling effective inverse problem solving by leveraging a trained decoder for sparse signal reconstruction.

## Contribution

It presents a new sparse synthesis regularization framework with neural networks that incorporates $ ext{l}^1$-penalty in training, differing from traditional frame-based methods.

## Key findings

- Decoder network enables sparse signal reconstruction with thresholded coefficients.
- The $ ext{l}^1$-Tikhonov functional acts as a regularization method for inverse problems.
- Proven effectiveness in reconstructing signals with sparse synthesis prior.

## Abstract

We propose a sparse reconstruction framework for solving inverse problems. Opposed to existing sparse regularization techniques that are based on frame representations, we train an encoder-decoder network by including an $\ell^1$-penalty. We demonstrate that the trained decoder network allows sparse signal reconstruction using thresholded encoded coefficients without losing much quality of the original image. Using the sparse synthesis prior, we propose minimizing the $\ell^1$-Tikhonov functional, which is the sum of a data fitting term and the $\ell^1$-norm of the synthesis coefficients, and show that it provides a regularization method.