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 phenomenon 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 phenomenon 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 result relies on showing that convolutional generators fit the structured part of an image significantly faster than the corrupted portion.

卷积神经网络在图像生成、还原和修复中具有良好的性能，即使在没有训练数据的情况下，通过应用随机初始化的过参数化凸轮生成器拟合噪声和损坏的自然图像可以去除图像噪声。在这篇论文中，作者将这种现象归因于卷积网络的特定架构选择，即使用固定插值滤波器进行卷积，并证明早期停止的梯度下降可用于去噪和对其他逆问题进行正则化。

利用卷积生成器的结构偏差进行降噪和正则化