${\ell }_p$-Power Heavy-Tailed Diffusion Models

We develop a variational framework for heavy-tailed diffusion based on an (\ell_p )-power family, extending Student- (t )-based models beyond (p=2 ). Using the family’s Gaussian scale-mixture structure, we construct diffusion paths for (p\in[1,2] ) via random-precision Gaussians and derive an adjacent-state $p$-th power objective through (\gamma )-power-divergence projection. At (p=2 ), the objectives are equivalent up to a constant, but differ for (p\in[1,2) ). Theoretical analysis identifies regimes where the adjacent-state objective is more sensitive to rare components than denoising, and conditions under which (p\in(1,2\) ) yields lower reconstruction risk than either endpoint. Experiments on synthetic data, language, and image generation reveal complementary strengths over Gaussian and Student- (t ) baselines: the adjacent-state objective better preserves rare modes under heavy-tailed noise, while denoising achieves better reconstruction on contaminated data.