Learning without relearning physical symmetry: an SO(3)-equivariant convolutional network (CNN) on the sphere

TL;DR

The paper introduces an SO(3)-equivariant CNN designed for spherical data using HEALPix pixelization, leveraging steerable filters, parallel transport, and spin-weighted features to achieve exact symmetry preservation. The model enables high-accuracy regression and denoising tasks on the sphere with minimal simulation overhead, significantly improving efficiency in simulation-based inference.

Why It Matters

In fields like cosmology, where high-fidelity simulations are costly and symmetry is intrinsic, this method reduces reliance on approximate models by embedding physical symmetries directly into the network. This leads to tighter inference posteriors and enables scalable, accurate neural analysis of spherical data without training-induced symmetry errors.

Key Findings

• The architecture achieves provably exact SO(3)-equivariance under band-limited conditions using steerable filters and spin-weighted features.
• On benchmark tasks, the model delivers state-of-the-art performance in dense gradient regression and cosmic microwave background foreground removal.
• In simulation-based inference for cosmic lensing, the method produces tighter posterior distributions than alternatives at the same simulation budget.