The Hessian perspective into the Nature of Convolutional Neural Networks

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 CNNs get manifested in their 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 in practice also hold for more general settings. Overall, our work generalizes and further establishes the key insight that the Hessian rank grows as the square root of the number of parameters, even in CNNs.