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

Imagine you're working on simulations in cosmology, where data is often represented on a sphere. Traditional methods can struggle with this, leading to inefficiencies and inaccuracies because they don't account for the sphere's inherent symmetry. This can result in simulations that require excessive computational resources, which is particularly problematic when high-fidelity simulations are needed for accurate scientific analysis. This issue is known as the inefficiency of non-equivariant methods, which can bias results and force reliance on less accurate models. To tackle this, the authors propose a spherical convolutional neural network (CNN) that respects the symmetry of spherical data. By using a gauge equivariant mesh architecture, they ensure that the network inherently understands the spherical structure without needing to learn it from the data. This architecture employs steerable filters and spin-weighted features to achieve exact equivariance, which significantly reduces simulation errors. The result is a method that not only enhances the accuracy of dense gradient regression but also excels in tasks like cosmic microwave background foreground removal and denoising. Compared to previous approaches, this new architecture allows for more efficient simulations, particularly in simulation-based inference where computational resources are limited. It enables tighter posteriors in cosmological lensing tasks, making it a valuable tool for researchers needing reliable analyses of spherical data.

The approach introduces a novel spherical architecture that directly addresses the symmetry of data on the sphere.

The method achieves provably exact equivariance and demonstrates strong performance on relevant benchmarks.