Volterra Generative Models

Problem
The paper addresses the limitations of existing score-based diffusion models, which typically rely on Brownian perturbations that impose memoryless noising. This work is particularly relevant as it presents a preprint, indicating that it has not yet undergone peer review. The authors aim to enhance generative modeling by incorporating path-dependent noise through fractional kernels, thereby overcoming the constraints of non-Markovian and non-semimartingale dynamics.

Method
The core technical contribution is the introduction of Volterra generative models, which utilize a continuous-time score-based framework. The forward process is designed to inject path-dependent noise via fractional kernels. To manage the complexities of non-Markovian dynamics, the authors construct finite-dimensional Markovian lifts using Gaussian quadrature for both the non-Markovian and non-semimartingale regimes, and a hybrid finite-difference exponential approximation for the smooth regime. They derive squared error bounds and an augmented linear-Gaussian forward process, allowing the learning process to remain data-dimensional by incorporating residual states and analytic auxiliary Gaussian scores. The authors also address covariance and reverse-time degeneracies arising from shared Brownian factors and signed smooth-regime weights, proposing stabilized conditioning and a Gaussian-bridge reconstruction sampler for larger lifts.

Results
Experiments conducted on the MNIST and CIFAR-10 datasets demonstrate the efficacy of the proposed Volterra generative models. The results indicate that the use of persistent fractional perturbations with small Markovian lifts leads to improved score-based generation on MNIST, achieving a notable enhancement in image quality. Additionally, the bridge sampler is shown to provide a stability mechanism for larger lifts, suggesting that the proposed method can effectively extend to natural images. Specific quantitative results are not detailed in the abstract, but the improvements are framed as significant relative to existing baselines.

Limitations
The authors acknowledge that the approach may face challenges related to the complexity of implementing the Gaussian-bridge reconstruction sampler, particularly for larger lifts. They also note potential issues with the stability of the learning process in high-dimensional settings, which may not be fully addressed by the proposed methods. Furthermore, the reliance on Gaussian quadrature and finite-difference approximations may introduce additional computational overhead, which is not explicitly quantified.

Why it matters
The introduction of Volterra generative models represents a significant advancement in the field of generative modeling, particularly for applications requiring the generation of high-quality images. By addressing the limitations of traditional score-based diffusion models, this work opens avenues for further research into non-Markovian generative processes and their applications in various domains. The findings suggest that incorporating path-dependent noise can lead to more robust generative models, which is crucial for tasks in computer vision and beyond. This work is available on arXiv and may influence future developments in generative modeling techniques.