BernNet: Learning Arbitrary Graph Spectral Filters via Bernstein Approximation

TL;DR

BernNet introduces a novel graph neural network that uses Bernstein polynomial approximation to learn arbitrary spectral filters, enabling more flexible and effective graph signal processing.

Contribution

The paper proposes BernNet, a new GNN framework that employs Bernstein polynomial approximation for designing and learning arbitrary spectral filters with theoretical support.

Findings

BernNet can learn complex spectral filters like band-rejection and comb filters.

It outperforms existing methods in real-world graph tasks.

The approach provides a principled way to design spectral filters with theoretical guarantees.

Abstract

Many representative graph neural networks, e.g., GPR-GNN and ChebNet, approximate graph convolutions with graph spectral filters. However, existing work either applies predefined filter weights or learns them without necessary constraints, which may lead to oversimplified or ill-posed filters. To overcome these issues, we propose BernNet, a novel graph neural network with theoretical support that provides a simple but effective scheme for designing and learning arbitrary graph spectral filters. In particular, for any filter over the normalized Laplacian spectrum of a graph, our BernNet estimates it by an order-$K$ Bernstein polynomial approximation and designs its spectral property by setting the coefficients of the Bernstein basis. Moreover, we can learn the coefficients (and the corresponding filter weights) based on observed graphs and their associated signals and thus achieve the BernNet specialized for the data. Our experiments demonstrate that BernNet can learn arbitrary spectral filters, including complicated band-rejection and comb filters, and it achieves superior performance in real-world graph modeling tasks. Code is available at https://github.com/ivam-he/BernNet.