  Classical results on the statistical complexity of linear models have
commonly identified the norm of the weights $\|w\|$ as a fundamental capacity
measure. Generalizations of this measure to the setting of deep networks have
been varied, though a frequently identified quantity is the product of weight
norms of each layer. In this work, we show that for a large class of networks
possessing a positive homogeneity property, similar bounds may be obtained
instead in terms of the norm of the product of weights. Our proof technique
generalizes a recently proposed sampling argument, which allows us to
demonstrate the existence of sparse approximants of positive homogeneous
networks. This yields covering number bounds, which can be converted to
generalization bounds for multi-class classification that are comparable to,
and in certain cases improve upon, existing results in the literature. Finally,
we investigate our sampling procedure empirically, which yields results
consistent with our theory.
