Polar Diffusion: Decoupling Radial and Tangential Denoising in Diffusion Models

Diffusion models typically perform denoising in Euclidean space using a single isotropic prediction objective, implicitly assigning equal importance to errors along all geometric directions. High-dimensional data, however, often concentrate near lower-dimensional structures, where radial and tangential variations can play fundamentally different roles in generation. We introduce Polar Diffusion, a geometric reformulation that factorizes the diffusion process over a product space of magnitude (radial) and normalized direction (tangential). We derive the corresponding product-space score-matching formulation and show that, when mapped back to Euclidean coordinates, the resulting dynamics recover the same marginal distributions as standard variance-preserving diffusion. This factorization exposes a degree of freedom hidden by the Euclidean formulation: radial and tangential denoising objectives can be controlled and weighted independently. Polar Diffusion thus provides a simple geometric extension of conventional diffusion models that enables explicit control over the relative contributions of magnitude and structural errors during denoising. Ablation studies show that moderately down-weighting the radial objective consistently improves generation quality, suggesting that radial and tangential errors need not contribute equally to effective denoising, with greater emphasis on the tangential component providing closer accounting for the distribution of structural information in the data.