Differentiable and accelerated spherical harmonic and Wigner transforms

TL;DR

This paper introduces highly efficient, differentiable algorithms for spherical harmonic and Wigner transforms, enabling fast, parallelized computations with gradients on modern hardware like GPUs, crucial for scientific and machine learning applications involving spherical data.

Contribution

The paper presents novel recursive algorithms for Wigner d-functions, coupled with separable transforms, enabling highly parallel, differentiable spherical harmonic and Wigner transforms with near-linear scaling on GPUs.

Findings

Achieves up to 400-fold acceleration over existing C implementations.

Supports various sampling schemes including equiangular and HEALPix.

Exhibits near-linear scaling with multiple GPUs, enabling efficient high-throughput computations.

Abstract

Many areas of science and engineering encounter data defined on spherical manifolds. Modelling and analysis of spherical data often necessitates spherical harmonic transforms, at high degrees, and increasingly requires efficient computation of gradients for machine learning or other differentiable programming tasks. We develop novel algorithmic structures for accelerated and differentiable computation of generalised Fourier transforms on the sphere $\mathbb{S}^2$ and rotation group $\text{SO}(3)$, i.e. spherical harmonic and Wigner transforms, respectively. We present a recursive algorithm for the calculation of Wigner $d$-functions that is both stable to high harmonic degrees and extremely parallelisable. By tightly coupling this with separable spherical transforms, we obtain algorithms that exhibit an extremely parallelisable structure that is well-suited for the high throughput computing of modern hardware accelerators (e.g. GPUs). We also develop a hybrid automatic and manual differentiation approach so that gradients can be computed efficiently. Our algorithms are implemented within the JAX differentiable programming framework in the S2FFT software code. Numerous samplings of the sphere are supported, including equiangular and HEALPix sampling. Computational errors are at the order of machine precision for spherical samplings that admit a sampling theorem. When benchmarked against alternative C codes we observe up to a 400-fold acceleration. Furthermore, when distributing over multiple GPUs we achieve very close to optimal linear scaling with increasing number of GPUs due to the highly parallelised and balanced nature of our algorithms. Provided access to sufficiently many GPUs our transforms thus exhibit an unprecedented effective linear time complexity.