On the Compressive Power of Deep Rectifier Networks for High Resolution Representation of Class Boundaries

This paper provides a theoretical justification of the superior classification performance of deep rectifier networks over shallow rectifier networks from the geometrical perspective of piecewise linear (PWL) classifier boundaries. We show that, for a given threshold on the approximation error, the required number of boundary facets to approximate a general smooth boundary grows exponentially with the dimension of the data, and thus the number of boundary facets, referred to as boundary resolution, of a PWL classifier is an important quality measure that can be used to estimate a lower bound on the classification errors. However, learning naively an exponentially large number of boundary facets requires the determination of an exponentially large number of parameters and also requires an exponentially large number of training patterns. To overcome this issue of "curse of dimensionality", compressive representations of high resolution classifier boundaries are required. To show the superior compressive power of deep rectifier networks over shallow rectifier networks, we prove that the maximum boundary resolution of a single hidden layer rectifier network classifier grows exponentially with the number of units when this number is smaller than the dimension of the patterns. When the number of units is larger than the dimension of the patterns, the growth rate is reduced to a polynomial order. Consequently, the capacity of generating a high resolution boundary will increase if the same large number of units are arranged in multiple layers instead of a single hidden layer. Taking high dimensional spherical boundaries as examples, we show how deep rectifier networks can utilize geometric symmetries to approximate a boundary with the same accuracy but with a significantly fewer number of parameters than single hidden layer nets.