Annealed L\'{e}vy Diffusion via Gaussian Surrogate Reverse Dynamics

Abstract

Gaussian diffusion models offer a practical framework for generative modeling. Denoising L\'{e}vy Probabilistic Models (DLPM) introduce heavy-tailed perturbations with a fixed tail index throughout denoising. We introduce Annealed L\'{e}vy Diffusion (ALD). ALD varies the stability index $\alpha(t)$ to combine heavy-tailed exploration with Gaussian refinement. Our marginal-first formulation represents stable noise as a scalar Gaussian scale mixture. This formulation enables standard Gaussian noise-prediction training with one additional scale input. We study two sampling processes. ALD-renoise estimates the clean sample and injects fresh marginal noise. ALD-posterior preserves dependence between successive states through an accumulated conditional variance and a Gaussian bridge update. Both samplers reuse the same trained predictor with different scalar inputs. We evaluate ALD on four 2D benchmarks and an imbalanced nine-mode mixture. We also examine training efficiency and the effects of prediction perturbations on clipped Holtsmark data. Additional experiments evaluate MNIST and CelebA64 generation and CIFAR-10LT augmentation. ALD preserves local geometry on the evaluated light-tailed and manifold-structured targets.