Convergence and Recovery Guarantees of Unsupervised Neural Networks for Inverse Problems

TL;DR

This paper provides theoretical convergence and recovery guarantees for unsupervised neural networks applied to inverse problems, bridging empirical success with rigorous analysis, and establishing overparametrization bounds for specific network architectures.

Contribution

It introduces deterministic guarantees for convergence and recovery in unsupervised neural networks solving inverse problems, connecting empirical methods with theoretical foundations.

Findings

Provides deterministic convergence guarantees.

Establishes recovery guarantees for neural networks.

Derives overparametrization bounds for two-layer Deep Inverse Prior networks.

Abstract

Neural networks have become a prominent approach to solve inverse problems in recent years. While a plethora of such methods was developed to solve inverse problems empirically, we are still lacking clear theoretical guarantees for these methods. On the other hand, many works proved convergence to optimal solutions of neural networks in a more general setting using overparametrization as a way to control the Neural Tangent Kernel. In this work we investigate how to bridge these two worlds and we provide deterministic convergence and recovery guarantees for the class of unsupervised feedforward multilayer neural networks trained to solve inverse problems. We also derive overparametrization bounds under which a two-layers Deep Inverse Prior network with smooth activation function will benefit from our guarantees.