JumpSampler: Making Distribution Tails Easier to Chase

Real-world data distributions can exhibit heavy-tailed characteristics in both pixel and latent spaces, assigning non-negligible probability mass to rare or low-density regions. However, diffusion and flow-based generative models rely predominantly on local, continuous sampling dynamics, which can be insufficient for capturing such heavy-tailed structures, potentially leading to under-represented tail events, reduced sample diversity, and distorted distributional statistics. In this paper, we provide a theoretical characterization of this limitation. Specifically, we show that, under standard regularity conditions, most samplers produce light-tailed terminal distributions, whereas sparse heavy-tailed perturbations can induce polynomial tail behavior while keeping the deviation from the original sampler controlled. Therefore, we design **JumpSampler**, a training-free and plug-in sampling strategy that augments pretrained diffusion and flow models with jumps, enabling occasional nonlocal transitions while remaining broadly compatible with diverse sampling schemes and generative architectures. The key design is to inject heavy-tailed perturbations only at sparse sampling steps and progressively anneal their effect over the sampling trajectory, allowing the sampler to explore distant low-density regions while preserving local refinement near the end of generation. We theoretically show that such sparse heavy-tailed jumps induce polynomial tail behavior in the resulting distribution, and further establish stability guarantees showing that the jump-augmented sampler remains controlled in both total variation and Wasserstein distance from the original sampling process. Extensive experiments demonstrate that JumpSampler improves tail fitting, distributional matching, and overall generation quality across both stochastic and deterministic samplers.