A library for differentiable signal processing and machine learning on the sphere

2026-09-30

AI Summary

This study addresses the lack of efficient machine learning tools for spherical data that simultaneously preserve topological and symmetry properties by introducing the torch-harmonics library. The proposed method leverages spherical harmonic transforms, vector spherical harmonics, and spectral convolutions, integrated with PyTorch's differentiable programming capabilities. It implements both discrete and continuous convolutions alongside global and local attention mechanisms, thereby establishing the core building blocks for Spherical Transformers. The primary contribution lies in providing a differentiable spherical signal processing and rotation-aware learning framework that supports scalable training and inference for scientific and engineering applications. Ultimately, this work effectively resolves critical bottlenecks in deep learning on spherical geometries.

Abstract

The two-dimensional sphere embedded in three-dimensional Euclidean space S2, plays a central role in a variety of scientific and engineering domains, including geophysics, planetary science, geodesy, atmospheric physics, quantum chemistry, cosmology, and virtual reality, among many others. As machine learning increasingly permeates these fields, the demand grows for robust tools that process and model functions on the sphere, while respecting the inherent topological and symmetry properties of the domain. We present torch-harmonics, a comprehensive library that offers efficient, differentiable implementations of advanced signal processing and machine learning (ML) methods for spherical data. These include the spherical harmonic transform (SHT), the spherical analogue of the Fourier transform, vector spherical harmonics, discrete-continuous and spectral convolutions, as well as both global and neighborhood spherical attention mechanisms. Beyond traditional representations, torch-harmonics provides the building blocks for state-of-the-art spherical ML architectures such as spherical transformers in order to enable scalable, rotationally-aware learning and inference in modern scientific and engineering applications.

Problem

• spherical signal processing
• differentiable machine learning
• spherical harmonics
• spherical data

Innovation

• differentiable spherical signal processing
• spherical harmonic transform
• spherical attention mechanisms
• spherical transformers
• rotationally-aware learning