The Hessian perspective into the Nature of Convolutional Neural Networks

TL;DR

This paper investigates the structure of convolutional neural networks through their Hessian maps, revealing that the Hessian rank grows proportionally to the square root of the number of parameters, providing new theoretical insights.

Contribution

It introduces a Toeplitz-based framework to analyze CNN Hessians and establishes tight upper bounds on Hessian rank, connecting architectural properties to Hessian structure.

Findings

Hessian rank in CNNs grows as the square root of parameters.

The Toeplitz representation effectively captures Hessian structure.

Theoretical bounds match empirical observations.

Abstract

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.