做生成模型的研究者会发现，PTL-Diffusion 用周期终端分布解决了流形结构丢失的痛点，在低维流形数据上效果明显，值得在点云或人脸生成任务上试试。
标准扩散模型通常使用单一高斯分布作为终端参考分布，这难以捕捉数据在低维流形上的结构。PTL-Diffusion 提出一种新的前向噪声过程，其终端分布不再是单一不变分布，而是一族周期性的高斯分布，从而将相位结构直接嵌入前向动力学。该方法推导了封闭形式的前向边缘分布、周期高斯终端族和显式高斯反向后验，并引入不变平均正则化项来耦合相位条件反向动力学。在环面、圆柱点云和 Olivetti 人脸数据集上的实验表明，PTL-Diffusion 在流形级别的分布匹配上优于标准 DDPM，减少了相位条件误差、特征空间协方差误差和最近邻流形距离。这项工作为结构化终端参考分布提供了有前景的方向。
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.