ReLU Deep Neural Networks from the Hierarchical Basis Perspective
This provides theoretical insights for researchers in machine learning and numerical analysis, but it is incremental as it builds on existing connections between DNNs and finite elements.
The paper connects ReLU deep neural networks to hierarchical basis methods in finite elements, showing they can approximate quadratic functions and explicitly reproduce linear finite element functions on 2D uniform meshes with two hidden layers.
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.