Spectral Diffusion Models on the Sphere

• URL: http://arxiv.org/abs/2601.20498v1
• Date: Wed, 28 Jan 2026 11:19:37 GMT
• Title: Spectral Diffusion Models on the Sphere
• Authors: Pierpaolo Brutti, Claudio Durastanti, Francesco Mari,
• Abstract summary: Diffusion models provide a principled framework for generative modeling via differential equations and time-reversed dynamics.<n>We develop a diffusion modeling framework defined directly on finite-dimensional harmonic representations of real-valued functions on the sphere.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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