APPROXIMATE EQUIVARIANCE NEEDLET CONVOLUTION

Anonymous

Keywords: Geometric Deep Learning, Spherical CNN, Needlet, Multi-scale and Multi-resolution

TL;DR: Multi-scale and multi-resolution framework for Spherical CNN.

Abstract: This paper develops a new spherical neural network framework based on needlet convolutions that can capture the multiscale information and rotation invariant feature of spherical data. The spherical needlets are a wavelet tight frame on the sphere $\mathbb{S}^2$ which can transform the data into a frequency domain with different scales: the low-pass and high-passes that extract approximate and detail information from the spherical signal. As the output signal of the spherical needlet convolution lies on the rotation group SO(3), we generalize the needlets to SO(3) and define SO(3) needlet convolution. Wavelet shrinkage is used as a nonlinear activation to reduce the redundancy in the needlet high-pass representation, which enhances the robustness of the neural network. The $\mathbb{S}^2$ needlet convolution can be connected with multiple SO(3) needlet convolution layers to form a Needlet Approximate Equivariance Spherical Neural Network, thus providing a powerful framework to distill the geometric equivariance feature and trainable multiresolution analyzer. Experimental results on quantum chemistry regression and gravitational wave parameter estimation show its great potential for solving scientific challenges.