  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.
