Spherical Brownian Diffusion for Language Modeling

Abstract

Continuous diffusion language models involve two coupled design choices: how to corrupt continuous token representations and how to invert that corruption at generation time. Recent work has explored language generation on the unit sphere through geodesic and von Mises--Fisher (vMF) paths, raising the question of which forward process to use on this geometry and how much that choice matters relative to inference. We introduce Spherical Brownian Diffusion, using Brownian motion as the canonical isotropic diffusion on the sphere. Its heat semigroup provides a principled forward process, an additive dimension-free noise clock, and an exact ancestral bridge, while a clean-token posterior learned with cross-entropy supports multiple samplers. To separate corruption from inference, we derive high-dimensional approximation rates between Brownian and moment-matched vMF corruption and use this relationship for controlled substitutions. On Sudoku, Brownian and moment-matched vMF give nearly identical results across four samplers and five budgets under otherwise identical settings. In contrast, with the corruption process fixed, sampler choice produces a 25.0-point spread at a sampler budget of 180 calls. Using the ancestral bridge, we split this gain into a memory effect from re-noising the state, present on Sudoku and GSM8K, and a token-selection effect on GSM8K and largest at small budgets. Across reasoning tasks, categorical refinement improves performance, whereas unconditional text generation benefits from stochastic updates that preserve diversity.