在复杂几何结构上生成数据，无需依赖复杂的曲面信息。
Riemannian Denoising Diffusion Probabilistic Models
- 基于投影机制，仅需函数值和一阶导数即可建模
- 在SO(10)和氨基酸构型空间上成功生成高维数据
- 适合无完整几何信息的复杂流形生成任务
我们提出黎曼去噪扩散概率模型（RDDPMs），用于学习欧氏空间子流形上由函数等值面定义的概率分布，涵盖多数实际应用中的流形。现有流形生成方法依赖大量几何信息（如测地线或拉普拉斯-贝尔特拉米算子的特征函数），因而受限于可获取这些信息的流形。相比之下，我们的方法基于投影方案，仅需评估定义子流形的函数值及其一阶导数，适用范围更广。我们在连续时间极限下提供了理论分析，揭示了RDDPM与流形上基于得分的生成模型之间的联系。该方法在已有数据集及新采样自高维流形（即SO(10)和固定二面角的丙氨酸二肽构型空间）的数据集上均表现出色。
原文摘要 · Abstract (English)
We propose Riemannian Denoising Diffusion Probabilistic Models (RDDPMs) for learning distributions on submanifolds of Euclidean space that are level sets of functions, including most of the manifolds relevant to applications. Existing methods for generative modeling on manifolds rely on substantial geometric information such as geodesic curves or eigenfunctions of the Laplace-Beltrami operator and, as a result, they are limited to manifolds where such information is available. In contrast, our method, built on a projection scheme, can be applied to more general manifolds, as it only requires being able to evaluate the value and the first order derivatives of the function that defines the submanifold. We provide a theoretical analysis of our method in the continuous-time limit, which elucidates the connection between our RDDPMs and score-based generative models on manifolds. The capability of our method is demonstrated on datasets from previous studies and on new datasets sampled from two high-dimensional manifolds, i.e. $\mathrm{SO}(10)$ and the configuration space of molecular system alanine dipeptide with fixed dihedral angle.
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