Approximate Equivariance SO(3) Needlet Convolution
This provides a powerful tool for scientific domains like quantum chemistry and cosmology that require handling high-resolution, multi-scale spherical signals with rotation invariance.
The paper tackled the problem of extracting rotation-invariant features from spherical signals by developing a Needlet approximate Equivariance Spherical CNN (NES) with SO(3) needlet convolutional layers, achieving state-of-the-art performance in quantum chemistry regression and Cosmic Microwave Background delensing reconstruction.
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.