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

arXiv: https://arxiv.org/abs/2606.09816 · HTML: https://mlsanity.com/paper/2606.09816/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models.
- Builds on 7 of its 29 references that are in the map.
- First posted on arXiv 8 Jun 2026.

## Placed in

- Diffusion models / [Diffusion foundations and sampling](https://mlsanity.com/diffusion/diffusion-foundations-and-sampling/)

## Builds on (most cited first, 7 of 7)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Diffusion Models Beat GANs on Image Synthesis](https://mlsanity.com/paper/2105.05233/) (2021)
- [Deep Unsupervised Learning using Nonequilibrium Thermodynamics](https://mlsanity.com/paper/1503.03585/) (2015)
- [Classifier-Free Diffusion Guidance](https://mlsanity.com/paper/2207.12598/) (2022)
- [Riemannian Score-Based Generative Modelling](https://mlsanity.com/paper/2202.02763/) (2022)
- [Riemannian Continuous Normalizing Flows](https://mlsanity.com/paper/2006.10605/) (2020)

## Abstract (arXiv)

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.