Volterra Generative Models

2026-06-16

AI Summary

This work addresses the limitations of conventional score-based diffusion models, which rely on memoryless Brownian motion noise and struggle to capture complex data structures. The authors propose a continuous-time generative framework that, for the first time, incorporates Volterra-type fractional noise into generative modeling. By employing a fractional kernel, the method introduces path-dependent perturbations and constructs a finite-dimensional Markovian lift to handle non-Markovian and non-semimartingale dynamics. A residual state formulation combined with an analytically tractable auxiliary Gaussian score mechanism ensures training stability while preserving the original data dimensionality, and reveals degeneracy issues caused by shared Brownian factors. The lifted process is built using Gaussian quadrature and hybrid finite-difference exponential approximations, and paired with a Gaussian bridge sampler and stable conditioning strategies. The approach outperforms baselines on MNIST and demonstrates scalability and sampling stability on high-dimensional natural images in CIFAR-10.

Abstract

Score-based diffusion models typically use Brownian perturbations, which provide tractable reverse-time dynamics but impose memoryless noising. We introduce Volterra generative models, a continuous-time score-based framework whose forward process injects path-dependent noise through fractional kernels. To handle the non-Markovian and non-semimartingale dynamics, we construct finite-dimensional Markovian lifts using Gaussian quadrature in both regimes and a hybrid finite-difference exponential approximation in the smooth regime. We prove squared error bounds, derive an augmented linear-Gaussian forward process, and show that the learning can remain data-dimensional by considering residual states and analytic auxiliary Gaussian scores. We also identify covariance and reverse-time degeneracies caused by shared Brownian factors and signed smooth-regime weights. The degeneracy motivates stabilized conditioning and, for stiff larger lifts, a Gaussian-bridge reconstruction sampler. Experiments on MNIST and CIFAR-10 show that persistent fractional perturbations with small Markovian lifts can improve score-based generation on MNIST and provide a promising extension to natural images, while the bridge sampler provides a stability mechanism for larger lifts.

Problem

• score-based generative models
• non-Markovian dynamics
• path-dependent noise
• fractional kernels
• diffusion models

Innovation

• Volterra generative models
• fractional kernels
• non-Markovian dynamics
• Markovian lifts
• Gaussian bridge sampler