ReLU Deep Neural Networks from the Hierarchical Basis Perspective

Abstract: We study ReLU deep neural networks (DNNs) by investigating their connections with the hierarchical basis method in finite element methods. First, we show that the approximation schemes of ReLU DNNs for $x^2$ and $xy$ are composition versions of the hierarchical basis approximation for these two functions. Based on this fact, we obtain a geometric interpretation and systematic proof for the approximation result of ReLU DNNs for polynomials, which plays an important role in a series of recent exponential approximation results of ReLU DNNs. Through our investigation of connections between ReLU DNNs and the hierarchical basis approximation for $x^2$ and $xy$, we show that ReLU DNNs with this special structure can be applied only to approximate quadratic functions. Furthermore, we obtain a concise representation to explicitly reproduce any linear finite element function on a two-dimensional uniform mesh by using ReLU DNNs with only two hidden layers.