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Convolutional Neural Networks (CNNs) have emerged as highly successful toolsfor image generation, recovery, and restoration. A major contributing factor tothis success is that convolutional networks impose strong prior assumptionsabout natural images. A surprising experiment that highlights thisarchitectural bias towards natural images is that one can remove noise andcorruptions from a natural image without using any training data, by simplyfitting (via gradient descent) a randomly initialized, over-parameterizedconvolutional generator to the corrupted image. While this over-parameterizednetwork can fit the corrupted image perfectly, surprisingly after a fewiterations of gradient descent it generates an almost uncorrupted image. Thisintriguing phenomenon enables state-of-the-art CNN-based denoising andregularization of other inverse problems. In this paper, we attribute thiseffect to a particular architectural choice of convolutional networks, namelyconvolutions with fixed interpolating filters. We then formally characterizethe dynamics of fitting a two-layer convolutional generator to a noisy signaland prove that early-stopped gradient descent denoises/regularizes. Our proofrelies on showing that convolutional generators fit the structured part of animage significantly faster than the corrupted portion.

Denoising and Regularization via Exploiting the Structural Bias of Convolutional Generators