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The Spectral Bias of Polynomial Neural Networks
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Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Add a multiplicative interaction layer | Two-layer ReLU network · Unit sphere, high-frequency spherical harmonics | NTK eigenvalue decay rate: Ω(k^{-d-1}) -> Ω(k^{-d/2-2}) | Helped |
| Use multiplicative layers instead of a standard feedforward network | Six-layer network · Sinusoidal signals, frequencies 5–50 | High-frequency learning speed | Helped |
| Use multiplicative layers rather than simply increasing network depth (ablation) | Sinusoid-learning network · Sinusoidal signals | High-frequency learning speed | Helped |
| Replace additive skip connections with multiplicative interactions (ablation) | Same-depth network · Sinusoidal signals | High-frequency learning speed | Helped |
| Add more multiplicative interactions (ablation) | Π-Net · High-frequency perturbation robustness | Robustness to perturbations | Helped |
| Use multiplicative interactions | U-Net in DIP denoising · Noisy natural images | Iterations to peak PSNR: approximately half the number of iterations | Helped |
| Use multiplicative interactions | U-Net in DIP denoising · Noisy natural images | Peak PSNR | No clear effect |
| Use multiplicative interactions | U-Net · Natural image power-spectrum matching | High-frequency learning speed | Helped |
| Add one multiplicative layer | Six-layer network · MNIST 3-vs-8 with high-frequency label noise | Validation-loss dip | Helped |
| Add one multiplicative layer | Six-layer network · MNIST 3-vs-8 without label noise | Validation loss | No clear effect |
| Increase the number of multiplicative layers (ablation) | Π-Net · MNIST 3-vs-8 with label noise | Validation-loss dip | Helped |
Closest papers by meaning
- How Universal Polynomial Bases Enhance Spectral Graph Neural Networks: Heterophily, Over-smoothing, and Over-squashing (ICML 2024 · similarity 0.88)
- Position: Spectral GNNs Rely Less on Graph Fourier Basis than Conceived (ICML 2025 · similarity 0.87)
- Identifiability of Deep Polynomial Neural Networks (NeurIPS 2025 · similarity 0.87)
- Learning Hierarchical Polynomials with Three-Layer Neural Networks (ICLR 2024 · similarity 0.86)
- Addressing Spectral Bias of Deep Neural Networks by Multi-Grade Deep Learning (NeurIPS 2024 · similarity 0.86)
- Improving Expressive Power of Spectral Graph Neural Networks with Eigenvalue Correction (AAAI 2024 · similarity 0.85)
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