PTL-Diffusion: Manifold-Aware Diffusion with Periodic Terminal Laws

2026-06-08

AI Summary

Standard diffusion models employ a single Gaussian terminal distribution, which struggles to effectively capture data structures residing on low-dimensional manifolds. This work proposes a novel diffusion framework that, for the first time, incorporates a family of periodic Gaussian terminal distributions into the forward dynamics via a forced Ornstein–Uhlenbeck process, yielding closed-form forward marginal distributions. In the reverse process, phase-conditional noise prediction and mean-invariance regularization are introduced to better model manifold geometry. The approach remains compatible with standard denoising training pipelines and demonstrates significant improvements over DDPM baselines on torus and cylindrical point clouds as well as the Olivetti faces dataset, consistently reducing phase-conditioning error, feature-space covariance error, and nearest-neighbor manifold distance.

Abstract

Standard diffusion models typically use a single time-homogeneous Gaussian terminal distribution as the reference law for generation. While this choice is analytically convenient and empirically powerful, it provides little explicit structure for data concentrated near low-dimensional manifolds, where different regions of the data distribution may correspond to distinct local geometric or semantic factors. As a result, the reverse model must recover manifold-level structure almost entirely from an unstructured terminal reference distribution. We propose PTL-Diffusion, a proof-of-concept diffusion framework whose forward noising process converges to a nonconstant periodic family of Gaussian terminal laws rather than to a single invariant law. Unlike a phase-conditioned DDPM, where phase information only enters the denoising network while the forward process remains unchanged, PTL-Diffusion embeds phase structure directly into the forward noising dynamics. The proposed construction remains close to standard denoising diffusion models: for a periodically forced Ornstein--Uhlenbeck-type forward process, we derive closed-form forward marginals, the limiting periodic Gaussian terminal family, and explicit Gaussian reverse posteriors, enabling standard noise-prediction training. We also introduce an invariant-average regularization term coupling the phase-conditioned reverse dynamics through the averaged periodic reference law. Experiments on torus and cylinder point-cloud benchmarks and the Olivetti face dataset show that PTL-Diffusion improves manifold-level distributional matching over matched DDPM baselines, reducing phase-conditioned errors, feature-space covariance errors, and nearest-neighbour manifold distances. These results suggest structured terminal reference laws as a promising direction, while motivating more expressive phase constructions and larger-scale evaluations.

Problem

• diffusion models
• manifold learning
• terminal distribution
• periodic structure
• generative modeling

Innovation

• Periodic Terminal Laws
• Manifold-Aware Diffusion
• Phase-Conditioned Dynamics
• Ornstein-Uhlenbeck Process
• Structured Reference Distribution