The Hessian perspective into the Nature of Convolutional Neural Networks
Sidak Pal Singh Thomas Hofmann Bernhard Schölkopf
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.