Compressive sensing with un-trained neural networks: Gradient descent finds the smoothest approximation

TL;DR

This paper demonstrates that un-trained convolutional neural networks can effectively recover structured signals and images from minimal measurements by leveraging their inherent self-regularization property during gradient descent.

Contribution

The paper provides numerical and theoretical evidence that un-trained CNNs can reconstruct signals from few measurements without explicit regularization, highlighting their self-regularizing capabilities.

Findings

Un-trained CNNs can recover natural images from limited measurements.

Gradient descent leads to the smoothest approximation fitting the data.

Un-trained CNNs require no additional regularization for successful reconstruction.

Abstract

Un-trained convolutional neural networks have emerged as highly successful tools for image recovery and restoration. They are capable of solving standard inverse problems such as denoising and compressive sensing with excellent results by simply fitting a neural network model to measurements from a single image or signal without the need for any additional training data. For some applications, this critically requires additional regularization in the form of early stopping the optimization. For signal recovery from a few measurements, however, un-trained convolutional networks have an intriguing self-regularizing property: Even though the network can perfectly fit any image, the network recovers a natural image from few measurements when trained with gradient descent until convergence. In this paper, we provide numerical evidence for this property and study it theoretically. We show that---without any further regularization---an un-trained convolutional neural network can approximately reconstruct signals and images that are sufficiently structured, from a near minimal number of random measurements.