Deep Learning for Inverse Problems: Bounds and Regularizers

TL;DR

This paper introduces a theoretical framework for deep learning in inverse problems, providing generalization bounds based on the Jacobian matrix and proposing regularization strategies that improve performance in tasks like image super-resolution.

Contribution

It offers a new generalization bound for deep networks in inverse problems and develops efficient regularization methods grounded in this theory.

Findings

Regularization strategies outperform standard weight decay in inverse problems.

Deep networks with Jacobian-based regularization show improved accuracy.

Numerical experiments validate the effectiveness of the proposed methods.

Abstract

Inverse problems arise in a number of domains such as medical imaging, remote sensing, and many more, relying on the use of advanced signal and image processing approaches -- such as sparsity-driven techniques -- to determine their solution. This paper instead studies the use of deep learning approaches to approximate the solution of inverse problems. In particular, the paper provides a new generalization bound, depending on key quantity associated with a deep neural network -- its Jacobian matrix -- that also leads to a number of computationally efficient regularization strategies applicable to inverse problems. The paper also tests the proposed regularization strategies in a number of inverse problems including image super-resolution ones. Our numerical results conducted on various datasets show that both fully connected and convolutional neural networks regularized using the regularization or proxy regularization strategies originating from our theory exhibit much better performance than deep networks regularized with standard approaches such as weight-decay.