论文标题
在随机卷积设计下的多层状态进化
Multi-layer State Evolution Under Random Convolutional Design
论文作者
论文摘要
生成神经网络先验下的信号恢复已成为统计推断和计算成像中的有希望的方向。然而，重建算法的理论分析是具有挑战性的。对于具有完全连接的层和高斯I.I.D.的生成先验权重，这是通过严格的状态进化来通过多层近似消息（ML-AMP）算法实现的。但是，实用的生成先验通常是卷积的，允许计算益处和归纳偏见，因此高斯I.I.D.体重假设非常有限。在本文中，我们克服了这一局限性，并确定了ML-AMP对随机卷积层的状态进化。我们特别证明，随机卷积层与高斯矩阵属于相同的普遍性类别。我们的证明技术具有独立的兴趣，因为它在编码理论中建立了卷积矩阵与空间耦合的传感矩阵之间的映射。
Signal recovery under generative neural network priors has emerged as a promising direction in statistical inference and computational imaging. Theoretical analysis of reconstruction algorithms under generative priors is, however, challenging. For generative priors with fully connected layers and Gaussian i.i.d. weights, this was achieved by the multi-layer approximate message (ML-AMP) algorithm via a rigorous state evolution. However, practical generative priors are typically convolutional, allowing for computational benefits and inductive biases, and so the Gaussian i.i.d. weight assumption is very limiting. In this paper, we overcome this limitation and establish the state evolution of ML-AMP for random convolutional layers. We prove in particular that random convolutional layers belong to the same universality class as Gaussian matrices. Our proof technique is of an independent interest as it establishes a mapping between convolutional matrices and spatially coupled sensing matrices used in coding theory.
