While Convolutional Neural Networks (CNNs) have long been investigated and applied, as well as theorized, we aim to provide a slightly different perspective into their nature -- through the perspective of their Hessian maps. The reason is that the loss Hessian captures the pairwise interaction of parameters and therefore forms a natural ground to probe how the architectural aspects of CNN get manifested in its structure and properties. We develop a framework relying on Toeplitz representation of CNNs, and then utilize it to reveal the Hessian structure and, in particular, its rank. We prove tight upper bounds (with linear activations), which closely follow the empirical trend of the Hessian rank and hold in practice in more general settings. Overall, our work generalizes and establishes the key insight that, even in CNNs, the Hessian rank grows as the square root of the number of parameters.

翻译：尽管卷积神经网络（CNNs）已被长期研究、应用并理论化，我们旨在通过其Hessian映射的视角，提供一种略有不同的视角来审视其本质。原因在于损失Hessian捕获了参数的成对交互作用，因此构成了探究CNN架构特征如何在其结构与性质中体现的自然基础。我们建立了一个基于CNN的Toeplitz表示的框架，并利用该框架揭示Hessian的结构，特别是其秩。我们证明了（在线性激活函数下）紧的上界，该上界密切遵循Hessian秩的经验趋势，并且在更一般的实际设置中同样成立。总体而言，我们的工作推广并确立了一个关键洞见：即使在CNNs中，Hessian秩也随参数数量的平方根增长。