Spectral Diffusion Models on the Sphere

Diffusion models provide a principled framework for generative modeling via stochastic differential equations and time-reversed dynamics. However, extension of spectral diffusion approaches to spherical data 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 generally no longer equivalent, even in the band-limited setting. We establish a quantitative relationship between the two objectives, showing that the geometry-induced covariance of the spectral noise gives rise to a distinct, geometry-dependent inductive bias. We also derive the corresponding forward and reverse diffusion equations and characterize the induced noise covariance.