PTL-Diffusion: Manifold-Aware Diffusion with Periodic Terminal Laws
2026-06-08 • Computer Vision and Pattern Recognition
Computer Vision and Pattern RecognitionArtificial Intelligence
AI summaryⓘ
The authors propose a new type of diffusion model called PTL-Diffusion that modifies the usual way noise is added in these models. Instead of using a single simple Gaussian distribution as the endpoint, their method uses a repeating family of Gaussian distributions arranged in a cycle, which better captures data that lies on curved shapes or manifolds. This helps the model recover underlying structural patterns more accurately. They tested PTL-Diffusion on several datasets and found it improves matching the data’s geometric structure compared to standard methods. Their work suggests new ways to design noise processes for better generation of complex data.
diffusion modelsGaussian distributionmanifoldOrnstein–Uhlenbeck processdenoising diffusion probabilistic models (DDPM)forward noising processreverse diffusionperiodic terminal lawspoint cloudphase conditioning
Authors
Danqi Zhuang, Jisui Huang, Xiaoyue Xi, Andrew Kiggins, Xiaojie Wang, Ke Chen, Yue Wu
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.