Denoising and Regularization via Exploiting the Structural Bias of Convolutional Generators

Denoising and Regularization via Exploiting the Structural Bias of Convolutional Generators Reinhard Heckel and Mahdi Soltanolkotabi † Dept. of Electrical and Computer Engineering, Technical University of Munich † Dept. of Electrical and Computer Engineering, University of Southern California November 1, 2019 Abstract Convolutional Neural Networks (CNNs) have emerged as highly successful tools for image generation, recovery, and restoration. This success is often attributed to large amounts of training data. However, recent experimental findings challenge this view and instead suggest that a major contributing factor to this success is that convolutional networks impose strong prior assumptions about natural images. A surprising experiment that highlights this architectural bias towards natural images is that one can remove noise and corruptions from a natural image without using any training data, by simply fitting (via gradient descent) a randomly initialized, over-parameterized convolutional generator to the single corrupted image. While this over-parameterized network can fit the corrupted image perfectly, surprisingly after a few iterations of gradient descent one obtains the uncorrupted image. This intriguing phenomena enables state-of-the-art CNN-based denoising and regularization of linear inverse problems such as compressive sensing. In this paper we take a step towards demystifying this experimental phenomena by attributing this effect to particular architectural choices of convolutional networks, namely convolutions with fixed interpolating filters. We then formally characterize the dynamics of fitting a two layer convolutional generator to a noisy signal and prove that early-stopped gradient descent denoises/regularizes. This results relies on showing that convolutional generators fit the structured part of an image significantly faster than the corrupted portion. 1 Introduction Convolutional neural networks are extremely popular for image generation. The majority of image generating networks is convolutional, ranging from Deep Convolutional Generative Adversarial Networks (DC-GANs) [Rad 15] to the U-Net [Ron 15]. It is well known that convolutional neural networks incorporate implicit assumption about the signals they generate, such as pixels that are close being related. This makes them particularly well suited for representing sets of images or modeling distributions of images. It is less known, however, that those prior assumptions build into the architecture are so strong that convolutional neural networks are useful even without ever being exposed to training data. The latter was first shown in the Deep Image Prior (DIP) paper [Uly 18]. Ulyanov et al. [Uly 18] observed that when'training' an standard convolutional auto-encoder such as the popular U-net [Ron 15] on a single noisy image and regularizing by early stopping, the network performs image restoration such as denoising with state-of-the-art performance.