Deep Convolutional Networks are Hierarchical Kernel Machines
Abstract
In i-theory a typical layer of a hierarchical architecture consists of HW modules pooling the dot products of the inputs to the layer with the transformations of a few templates under a group. Such layers include as special cases the convolutional layers of Deep Convolutional Networks (DCNs) as well as the non-convolutional layers (when the group contains only the identity). Rectifying nonlinearities -- which are used by present-day DCNs -- are one of the several nonlinearities admitted by i-theory for the HW module. We discuss here the equivalence between group averages of linear combinations of rectifying nonlinearities and an associated kernel. This property implies that present-day DCNs can be exactly equivalent to a hierarchy of kernel machines with pooling and non-pooling layers. Finally, we describe a conjecture for theoretically understanding hierarchies of such modules. A main consequence of the conjecture is that hierarchies of trained HW modules minimize memory requirements while computing a selective and invariant representation.
Cite
@article{arxiv.1508.01084,
title = {Deep Convolutional Networks are Hierarchical Kernel Machines},
author = {Fabio Anselmi and Lorenzo Rosasco and Cheston Tan and Tomaso Poggio},
journal= {arXiv preprint arXiv:1508.01084},
year = {2015}
}