Riemannian Denoising Diffusion Probabilistic Models
Zichen Liu, Wei Zhang, Christof Sch\"utte, Tiejun Li
TL;DR
This paper introduces Riemannian Denoising Diffusion Probabilistic Models (RDDPMs) that can learn distributions on a wide class of manifolds using only function evaluations and derivatives, expanding generative modeling capabilities.
Contribution
The paper presents a projection-based RDDPM approach that applies to general manifolds without requiring geometric information like geodesics or eigenfunctions, supported by theoretical analysis.
Findings
Successfully modeled distributions on high-dimensional manifolds such as SO(10)
Demonstrated applicability to molecular configuration spaces
Provided theoretical insights linking RDDPMs to score-based models on manifolds
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…
Peer Reviews
Decision·Submitted to ICLR 2025
While I am not an expert on diffusion models, the work appears like a natural extension to the submanifold setting. The paper is easy to read and the experimental section seems thorough enough as it contains several examples from different domains. Furthermore, the accompanied code appears examplary and contains clear instructions on how to reproduce the results. Overall, this appears to be high-quality work.
In my opinion there is only one small missing element that is easy to address: the paper should contain more details regarding the computational cost of the proposed procedure. At the very least I would like to see runtimes for the experiments in the paper and a short appendix section explaining all the computational costs associated with the procedure, how they depend on the dimension of the sub-manifold, the complexity of the level-set function, etc. so that readers would have a good idea when
1. The paper is well-written and well-organized, making it easy to follow. 2. While the idea of using a projection scheme instead of relying on extensive manifold information, such as geodesics and Laplacian eigenfunctions, has been explored in previous works, it remains an interesting approach for defining a noising/denoising Markov chain with (implicitly given) transition kernel. 3. The authors conduct extensive experiments to demonstrate the effectiveness of the proposed model.
1. One of my main concerns is Equation (8), which defines the transition kernel. Although the map $G_x: \mathcal{M} \to T_x \mathcal{M}$ is well-defined, it is not a true bijection due to the possibility of multiple solutions arising from the constraint in Equation (5). This implies that Equation (8) may not hold in general. While the authors mention in a footnote that $\sigma$ can be chosen small enough to ensure that Equation (5) has at least some solution with high probability, the fundamenta
The techniques in this paper allows for training diffusion models on a more general class of manifolds than other methods, only requiring knowing the defining equation of the manifold. It solves a system of equations to ensure the points generated by the forward and reverse processes stays on the manifold, instead of requiring knowledge of the closest-point projection onto the manifold. There are extensive experiments on a variety of manifolds and datasets.
In my opinion, the disadvantages of this method (outlined below) outweighs the main advantages (allowing for more general manifolds). The main contribution seems to be solving for the projection numerically, which is only useful when the exact projection onto the manifold is not known, but there needs to be more evaluation on this front. 1. During training, the entire trajectory for the random walk needs to be generated. The training objective predicts the expected value of $x^{k}$ given $x^{
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
Taxonomy
TopicsMorphological variations and asymmetry · Gaussian Processes and Bayesian Inference · Generative Adversarial Networks and Image Synthesis
MethodsDiffusion