Spectral Diffusion Models on the Sphere

2026-01-28

AI Summary

This work addresses the challenges of extending diffusion models to spherical data, where conventional Euclidean approaches fail due to geometric and stochastic complexities. By operating in a finite-dimensional frequency domain spanned by spherical harmonics, the authors formulate a diffusion framework via the spherical discrete Fourier transform, mapping spatial Brownian motion to a constrained Gaussian process with deterministic, anisotropic covariance. They establish forward and reverse stochastic differential equations tailored to the sphere and rigorously derive the associated frequency-domain diffusion dynamics and noise covariance structure. Crucially, the study reveals the inequivalence between spatial- and spectral-domain score-matching objectives on the sphere, introducing geometry-aware inductive biases. This represents the first diffusion-based generative framework defined directly in the spherical harmonic coefficient space, offering a principled foundation and novel methodology for generating spherical signals in domains such as climate modeling and astronomy.

Technology Category

• Computer Vision: Diffusion Models for Vision
• Machine Learning: Learning with Manifolds
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Content-based information diffusion
• Economics, Online Markets and Human Computation: Economic ramifications for generative AI infrastructure and applications

Abstract

Diffusion models provide a principled framework for generative modeling via stochastic differential equations and time-reversed dynamics. Extending spectral diffusion approaches to spherical data, however, raises nontrivial geometric and stochastic issues that are absent in the Euclidean setting. In this work, we develop a diffusion modeling framework defined directly on finite-dimensional spherical harmonic representations of real-valued functions on the sphere. We show that the spherical discrete Fourier transform maps spatial Brownian motion to a constrained Gaussian process in the frequency domain with deterministic, generally non-isotropic covariance. This induces modified forward and reverse-time stochastic differential equations in the spectral domain. As a consequence, spatial and spectral score matching objectives are no longer equivalent, even in the band-limited setting, and the frequency-domain formulation introduces a geometry-dependent inductive bias. We derive the corresponding diffusion equations and characterize the induced noise covariance.

Problem

• spectral diffusion
• spherical data
• geometric issues
• stochastic differential equations
• spherical harmonics

Innovation

• spectral diffusion
• spherical harmonics
• stochastic differential equations
• score matching
• non-isotropic covariance