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.

翻译：扩散模型通过随机微分方程与时间反向动力学为生成建模提供了原则性框架。然而，将谱扩散方法扩展至球面数据时，会引发欧几里得设定中不存在的非平凡几何与随机问题。本研究开发了一种直接定义于球面实值函数有限维球谐函数表示的扩散建模框架。我们证明：球面离散傅里叶变换将空间布朗运动映射为频域中具有确定性且通常非各向同性协方差的约束高斯过程。这诱导出谱域中修正的前向与反向时间随机微分方程。其结果是，即使在带限设定下，空间与谱分数匹配目标也不再等价，且频域公式引入了依赖于几何的归纳偏置。我们推导了相应的扩散方程，并刻画了诱导噪声协方差的特性。