Spectral Diffusion Models on the Sphere

Claudio Durastanti and Francesco Mari present a diffusion modeling framework for spherical data using finite-dimensional spherical harmonic representations. The spherical discrete Fourier transform maps spatial Brownian motion to a constrained Gaussian process in the frequency domain with non-isotropic covariance, inducing modified forward- and reverse-time stochastic differential equations. The authors show that spatial and spectral score matching objectives are generally not equivalent, establishing a quantitative relationship between them and characterizing the geometry-dependent noise covariance.