BernNet: Learning Arbitrary Graph Spectral Filters via Bernstein Approximation (2106.10994v3)

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.

• The paper presents a novel GNN model using Bernstein polynomial approximations to learn arbitrary graph spectral filters, addressing limitations of fixed and unconstrained methods.
• It leverages non-negative Bernstein coefficients to ensure valid, interpretable filter design over the normalized Laplacian spectrum.
• Empirical results demonstrate BernNet's superior performance over standard models like GCN and ChebNet in complex graph signal processing tasks.

Overview of BernNet: Learning Arbitrary Graph Spectral Filters via Bernstein Approximation

The paper introduces BernNet, a novel graph neural network (GNN) model utilizing Bernstein polynomial approximation to design and learn arbitrary graph spectral filters. Traditional GNN models either rely on predefined filter weights or learn filter weights without constraints, potentially leading to suboptimal or poorly conditioned filters. BernNet addresses these limitations by offering a theoretically grounded method for constructing custom spectral filters tailored for graph-structured data.

Key Contributions

BernNet leverages the Bernstein polynomial for approximating filter functions over the normalized Laplacian spectrum of a graph. This approach provides several advantages:

1. Arbitrary Filter Approximation: BernNet can approximate any continuous filter function by leveraging Bernstein polynomial approximations of order $K$. This flexibility allows for constructing sophisticated filters like band-rejection and comb filters, which are challenging for existing GNN architectures.
2. Non-negative Bernstein Coefficients: The non-negative nature of Bernstein coefficients ensures that the learned filter is valid over the spectral range, adhering to non-negativity constraints on filter responses.
3. Interpretability and Simplicity: The choice of Bernstein basis functions facilitates intuitive filter design, as coefficients directly correlate with sampled filter values, enhancing the interpretability of the spectral filters realized.
4. Experimental Validation: Empirical results demonstrate BernNet’s ability to achieve superior performance in graph signal processing tasks, surpassing models like GCN, ChebNet, and GPR-GNN, particularly in learning complex spectral filters from data.

Theoretical Implications

The paper positions BernNet within a broader theoretical context in graph optimization. It establishes that any polynomial filter satisfying non-negative constraints must align with Bernstein polynomial representations. This effectively ensures that the filters designed or learned through BernNet adhere to the constraints required for deriving valid and interpretable spectral responses.

The exploration into spectral graph theory highlights the appropriateness of Bernstein polynomials for designing spectral filters, positing that arbitrary valid polynomial filters inherently correspond to the architecture proposed by BernNet.

Future Directions

The authors suggest that future research could explore extending BernNet to handle hypergraphs and dynamic graphs, where conventional GNN models may struggle with the complex spectral characteristics inherent to such structures. Additionally, there is potential for integrating BernNet with spatial GNN methods, aiming for hybrid models that can balance expressive power and computational efficiency.

Conclusion

BernNet effectively bridges the gap between fixed and unconstrained learning of graph spectral filters by providing a robust, interpretable framework grounded in Bernstein polynomial approximation. The model's ability to design and learn complex filters potentially unlocks new capabilities in graph signal processing applications, influencing both practical applications and theoretical advancements in spectral-based GNNs.