ReLU Deep Neural Networks from the Hierarchical Basis Perspective

TL;DR

This paper explores the connection between ReLU deep neural networks and hierarchical basis methods, providing geometric insights and explicit constructions for approximating polynomials and finite element functions.

Contribution

It reveals the hierarchical basis perspective of ReLU DNNs, offering new geometric interpretations and explicit representations for finite element functions.

Findings

ReLU DNNs approximate quadratic functions via hierarchical basis schemes.

Explicit ReLU DNN constructions reproduce linear finite element functions.

Connection established between ReLU DNNs and hierarchical basis approximation.

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.