Score-based Generative Models with Lévy Processes

Investigating the optimal stochastic process beyond Gaussian for noise injection in a score-based generative model remains an open question. Brownian motion is a light-tailed process with continuous paths, which leads to a slow convergence rate for the Number of Function Evaluation (NFE). Recent studies have shown that diffusion models suffer from mode-collapse issues on imbalanced data.In order to overcome the limitations of Brownian motion, we introduce a novel score-based generative model referred to as Lévy-Itō Model (LIM). This model utilizes isotropic $\alpha$-stable Lévy processes. We first derive an exact reverse-time stochastic differential equation driven by the Lévy process and develop the corresponding fractional denoising score matching. The proposed generative model takes advantage of the heavy-tailed properties of the Lévy process. Our experimental results show LIM allows for faster and more diverse sampling while maintaining high fidelity compared to existing diffusion models across various image datasets such as CIFAR10, CelebA, and imbalanced dataset CIFAR10LT. Comparing our results to those of DDPM with 3.21 Fréchet Inception Distance (FID) and 0.6437 Recall on the CelebA dataset, we achieve 1.58 FID and 0.7006 Recall using the same architecture. LIM shows the best performance in NFE 500 with $2\times$ faster total wall-clock time than the baseline.