Convolutional Neural Networks Analyzed via Inverse Problem Theory and Sparse Representations

TL;DR

This paper provides a theoretical analysis of CNNs for inverse imaging problems, demonstrating how they learn optimal solutions as filters and the importance of mutual coherence for convergence.

Contribution

It offers a mathematical validation of CNN training dynamics for inverse problems, linking residual learning and skip connections to mutual coherence and convergence.

Findings

CNN filters solve inverse problems during training

Mutual coherence is crucial for CNN convergence

Residual learning and skip connections enhance coherence and performance

Abstract

Inverse problems in imaging such as denoising, deblurring, superresolution (SR) have been addressed for many decades. In recent years, convolutional neural networks (CNNs) have been widely used for many inverse problem areas. Although their indisputable success, CNNs are not mathematically validated as to how and what they learn. In this paper, we prove that during training, CNN elements solve for inverse problems which are optimum solutions stored as CNN neuron filters. We discuss the necessity of mutual coherence between CNN layer elements in order for a network to converge to the optimum solution. We prove that required mutual coherence can be provided by the usage of residual learning and skip connections. We have set rules over training sets and depth of networks for better convergence, i.e. performance.