PowerNet: Efficient Representations of Polynomials and Smooth Functions by Deep Neural Networks with Rectified Power Units

TL;DR

This paper introduces PowerNets, a new class of neural networks with rectified power units that can exactly represent polynomials and approximate smooth functions more effectively than ReLU networks, especially in high-accuracy and smoothness-required scenarios.

Contribution

The paper develops explicit algorithms for constructing PowerNets with sparse RePU connections, enabling exact polynomial representation and improved approximation of smooth functions.

Findings

PowerNets can exactly represent polynomials up to degree s^n with n layers.

The approximation error is bounded by the best polynomial approximation error.

PowerNets outperform ReLU networks in approximating smooth functions with high accuracy.

Abstract

Deep neural network with rectified linear units (ReLU) is getting more and more popular recently. However, the derivatives of the function represented by a ReLU network are not continuous, which limit the usage of ReLU network to situations only when smoothness is not required. In this paper, we construct deep neural networks with rectified power units (RePU), which can give better approximations for smooth functions. Optimal algorithms are proposed to explicitly build neural networks with sparsely connected RePUs, which we call PowerNets, to represent polynomials with no approximation error. For general smooth functions, we first project the function to their polynomial approximations, then use the proposed algorithms to construct corresponding PowerNets. Thus, the error of best polynomial approximation provides an upper bound of the best RePU network approximation error. For smooth functions in higher dimensional Sobolev spaces, we use fast spectral transforms for tensor-product grid and sparse grid discretization to get polynomial approximations. Our constructive algorithms show clearly a close connection between spectral methods and deep neural networks: a PowerNet with $n$ layers can exactly represent polynomials up to degree $s^n$, where $s$ is the power of RePUs. The proposed PowerNets have potential applications in the situations where high-accuracy is desired or smoothness is required.