# Harmonic Decompositions of Convolutional Networks

arXiv: https://arxiv.org/abs/2003.12756 · HTML: https://mlsanity.com/paper/2003.12756/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 28 Mar 2020.

## Built on by (most cited first, 6 of 6)

- [Approximation and Learning with Deep Convolutional Models: a Kernel Perspective](https://mlsanity.com/paper/2102.10032/) (2021)
- [Learning with convolution and pooling operations in kernel methods](https://mlsanity.com/paper/2111.08308/) (2021)
- [Eigenspace Restructuring: a Principle of Space and Frequency in Neural Networks](https://mlsanity.com/paper/2112.05611/) (2021)
- [U-Nets as Belief Propagation: Efficient Classification, Denoising, and Diffusion in Generative Hierarchical Models](https://mlsanity.com/paper/2404.18444/) (2024)
- [On the Spectral Bias of Convolutional Neural Tangent and Gaussian Process Kernels](https://mlsanity.com/paper/2203.09255/) (2022)
- [On the Shift Invariance of Max Pooling Feature Maps in Convolutional Neural Networks](https://mlsanity.com/paper/2209.11740/) (2022)

## Abstract (arXiv)

We present a description of the function space and the smoothness class associated with a convolutional network using the machinery of reproducing kernel Hilbert spaces. We show that the mapping associated with a convolutional network expands into a sum involving elementary functions akin to spherical harmonics. This functional decomposition can be related to the functional ANOVA decomposition in nonparametric statistics. Building off our functional characterization of convolutional networks, we obtain statistical bounds highlighting an interesting trade-off between the approximation error and the estimation error.