The Hessian perspective into the Nature of Convolutional Neural Networks
This provides a theoretical insight into CNN architecture for researchers, but it is incremental as it extends known Hessian analysis to CNNs.
The paper tackles the problem of understanding the nature of Convolutional Neural Networks (CNNs) by analyzing their Hessian maps, which capture parameter interactions, and proves that the Hessian rank grows as the square root of the number of parameters, with tight upper bounds that align with empirical trends.
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.