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[2505.04338] Riemannian Denoising Diffusion Probabilistic Models
Summary
The paper introduces Riemannian Denoising Diffusion Probabilistic Models (RDDPMs), which enhance generative modeling on submanifolds of Euclidean space, overcoming limitations of existing methods that require extensive geometric information.
Why It Matters
RDDPMs provide a novel approach to generative modeling that can be applied to a wider range of manifolds without needing detailed geometric data. This advancement could significantly impact fields such as computer vision and data science, where modeling complex data distributions is crucial.
Key Takeaways
- RDDPMs can model distributions on submanifolds without extensive geometric information.
- The method relies on evaluating function values and first-order derivatives, making it versatile.
- Theoretical analysis connects RDDPMs to score-based generative models.
- Demonstrated effectiveness on high-dimensional manifolds and molecular datasets.
- Potential applications in various fields, including computer vision and molecular modeling.
Computer Science > Machine Learning arXiv:2505.04338 (cs) [Submitted on 7 May 2025 (v1), last revised 14 Feb 2026 (this version, v2)] Title:Riemannian Denoising Diffusion Probabilistic Models Authors:Zichen Liu, Wei Zhang, Christof Schütte, Tiejun Li View a PDF of the paper titled Riemannian Denoising Diffusion Probabilistic Models, by Zichen Liu and 3 other authors View PDF HTML (experimental) Abstract: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 molec...
