The Spectral Bias of Polynomial Neural Networks

TL;DR

This paper analyzes the spectral bias of polynomial neural networks, revealing that certain parametrizations can accelerate learning of high-frequency components, which is crucial for tasks like image generation.

Contribution

The study provides a spectral analysis of PNNs' NTK, showing how the -Net family speeds up high-frequency learning, supported by extensive experiments.

Findings

-Net speeds up high-frequency learning

Spectral bias can be modulated by architecture design

Experimental results confirm theoretical predictions

Abstract

Polynomial neural networks (PNNs) have been recently shown to be particularly effective at image generation and face recognition, where high-frequency information is critical. Previous studies have revealed that neural networks demonstrate a $\textit{spectral bias}$ towards low-frequency functions, which yields faster learning of low-frequency components during training. Inspired by such studies, we conduct a spectral analysis of the Neural Tangent Kernel (NTK) of PNNs. We find that the $\Pi$-Net family, i.e., a recently proposed parametrization of PNNs, speeds up the learning of the higher frequencies. We verify the theoretical bias through extensive experiments. We expect our analysis to provide novel insights into designing architectures and learning frameworks by incorporating multiplicative interactions via polynomials.