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

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.