Approximate Equivariance SO(3) Needlet Convolution

TL;DR

This paper introduces a rotation-invariant needlet convolution for SO(3) that enables multiscale spherical signal analysis, leading to a new spherical CNN with state-of-the-art results in scientific applications.

Contribution

It generalizes spherical needlet transforms to SO(3) and develops a scalable, rotation-invariant spherical CNN with wavelet-based signal embedding.

Findings

Achieves state-of-the-art performance in quantum chemistry regression.

Effective in Cosmic Microwave Background delensing reconstruction.

Provides a scalable multi-resolution spherical signal analysis tool.

Abstract

This paper develops a rotation-invariant needlet convolution for rotation group SO(3) to distill multiscale information of spherical signals. The spherical needlet transform is generalized from $\mathbb{S}^2$ onto the SO(3) group, which decomposes a spherical signal to approximate and detailed spectral coefficients by a set of tight framelet operators. The spherical signal during the decomposition and reconstruction achieves rotation invariance. Based on needlet transforms, we form a Needlet approximate Equivariance Spherical CNN (NES) with multiple SO(3) needlet convolutional layers. The network establishes a powerful tool to extract geometric-invariant features of spherical signals. The model allows sufficient network scalability with multi-resolution representation. A robust signal embedding is learned with wavelet shrinkage activation function, which filters out redundant high-pass representation while maintaining approximate rotation invariance. The NES achieves state-of-the-art performance for quantum chemistry regression and Cosmic Microwave Background (CMB) delensing reconstruction, which shows great potential for solving scientific challenges with high-resolution and multi-scale spherical signal representation.