Spectral Diffusion Models on the Sphere

TL;DR

This paper extends spectral diffusion models to spherical data by developing a framework on spherical harmonic representations, addressing unique geometric challenges and deriving new diffusion equations for spherical functions.

Contribution

It introduces a novel spectral diffusion framework on the sphere, handling geometric complexities and formulating diffusion equations in the spectral domain.

Findings

Spectral diffusion on the sphere involves non-isotropic covariance in the frequency domain.

Spatial and spectral score matching objectives differ in the spherical setting.

The framework characterizes diffusion equations and noise covariance specific to spherical data.

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.