# Differentiable and accelerated spherical harmonic and Wigner transforms

arXiv: https://arxiv.org/abs/2311.14670 · HTML: https://mlsanity.com/paper/2311.14670/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 12 of its 78 references that are in the map.
- First posted on arXiv 24 Nov 2023.

## Builds on (most cited first, 12 of 12)

- [Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges](https://mlsanity.com/paper/2104.13478/) (2021)
- [Learning SO(3) Equivariant Representations with Spherical CNNs](https://mlsanity.com/paper/1711.06721/) (2017)
- [Clebsch-Gordan Nets: a Fully Fourier Space Spherical Convolutional Neural Network](https://mlsanity.com/paper/1806.09231/) (2018)
- [Efficient Generalized Spherical CNNs](https://mlsanity.com/paper/2010.11661/) (2020)
- [Scalable and Equivariant Spherical CNNs by Discrete-Continuous (DISCO) Convolutions](https://mlsanity.com/paper/2209.13603/) (2022)
- [Convolutional Networks for Spherical Signals](https://mlsanity.com/paper/1709.04893/) (2017)
- [Scattering Networks on the Sphere for Scalable and Rotationally Equivariant Spherical CNNs](https://mlsanity.com/paper/2102.02828/) (2021)
- [How can spherical CNNs benefit ML-based diffusion MRI parameter estimation?](https://mlsanity.com/paper/2207.00572/) (2022)
- [CosmoPower-JAX: high-dimensional Bayesian inference with differentiable cosmological emulators](https://mlsanity.com/paper/2305.06347/) (2023)
- [Proximal nested sampling with data-driven priors for physical scientists](https://mlsanity.com/paper/2307.00056/) (2023)
- [Rethinking the modeling of the instrumental response of telescopes with a differentiable optical model](https://mlsanity.com/paper/2111.12541/) (2021)
- [Learned Interferometric Imaging for the SPIDER Instrument](https://mlsanity.com/paper/2301.10260/) (2023)

## Built on by (most cited first, 6 of 6)

- [PRIMU: Uncertainty Estimation for Novel Views in Gaussian Splatting from Primitive-Based Representations of Error and Coverage](https://mlsanity.com/paper/2508.02443/) (2025)
- [Equivariant Neural Tangent Kernels](https://mlsanity.com/paper/2406.06504/) (2024)
- [Efficient fine-tuning of 37-level GraphCast with the Canadian global deterministic analysis](https://mlsanity.com/paper/2408.14587/) (2024)
- [3D Equivariant Pose Regression via Direct Wigner-D Harmonics Prediction](https://mlsanity.com/paper/2411.00543/) (2024)
- [Integral Formulas for Vector Signal Tensor Products](https://mlsanity.com/paper/2603.08630/) (2026)
- [Herglotz-NET: Implicit Neural Representation of Spherical Data with Harmonic Positional Encoding](https://mlsanity.com/paper/2502.13777/) (2025)

## Abstract (arXiv)

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 mathbbS2 and rotation group 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.