Volterra Generative Models via Markovian Lifting

Abstract

Diffusion-based generative models usually perturb data through Markov processes driven by Brownian motion. Volterra processes allow more flexible temporal dependence, but their non-Markovian structure complicates score learning and reverse-time simulation. We develop Volterra generative models and provide Markovian approximations. Gaussian conditioning then reduces the denoising problem to the original data dimension. We present the connections with variance-exploding and variance-preserving representations. Our framework also permits reuse of pretrained denoisers such as EDM2 networks. With one auxiliary factor, our models trained from scratch achieve FIDs of $0.547$ on MNIST and $1.860$ on conditional CIFAR-10. With autoguidance, ImageNet64 FID reaches $1.912$ after fine-tuning, while ImageNet512 FID reaches $1.903$ using frozen pretrained networks.

arXiv commentSubstantially revised manuscript; author list updated; 35 pages