Equivariant cnn improves spherical data analysis and inference efficiency
Learning without relearning physical symmetry: an SO(3)-equivariant convolutional network (CNN) on the sphere
Machine Learning
Summary
Datasets in many physical sciences come in shapes like spheres and have no natural direction, so analyzing them takes more effort if the analysis method ignores this. The authors created a new kind of neural network that respects the sphere's symmetries exactly, so it doesn't have to relearn these during training. This approach reduces the amount of simulation needed and improves accuracy in tasks like cosmic microwave background data cleaning. Their method especially helps when simulations are expensive by producing better results with fewer examples.
What this means in practice
- •For cosmology data teams: Produce tighter parameter estimates in lensing studies with fewer expensive simulations by using SO(3)-equivariant neural networks.
- •For satellite image analysts: Improve denoising and feature extraction from spherical satellite data by applying exact symmetry-respecting convolutions.
Authors
Joshua Williamson, Niall Jeffrey, Lorne Whiteway
Abstract
Many of the most important datasets in the physical sciences live natively on the sphere, and these data often carry no preferred position or orientation. Using methods that do not reflect this symmetry leads to far higher simulation requirements, which can be catastrophic for scientific analyses (forcing reliance on cheap, imperfect models that bias results). We present a novel spherical implementation of a gauge equivariant mesh architecture that is specifically designed for the Hierarchical Equal Area isoLatitude Pixelization (\healpix{}) of the sphere. The $\mathrm{SO}(3)$-equivariant convolution is built from steerable filters, parallel transport and spin-weighted features. The architecture supplies the exact symmetry, so the network does not spend training data learning it. The convolution operations are constructed to give provably exact equivariance, given a band-limit. Our full CNNs can achieve SO(3)-invariance error at per-cent-level and lower, which is controlled through well-understood hyperparameters. On benchmarks, our technique gives the most accurate dense gradient regression and competitive cosmic microwave background foreground removal and denoising. The improved simulation efficiency matters most in simulation-based inference (SBI), where high-fidelity simulations are expensive and neural compressors require large training sets, so a fixed budget forces analyses with less accurate simulators. In cosmology lensing SBI, our technique yields the tightest posteriors at matched simulation count. This approach is a novel method that will enable new convolutional neural network analyses, including generative and inference tasks, for processes and data on the sphere.