Riemannian Score-Based Generative Modeling

By Valentin De Bortoli, Emile Mathieu, Michael F. Hutchinson, James Thornton, Yee Whye Teh, Arnaud Doucet. 2022

278 citations recorded by the source corpus.

Score-based generative models (SGMs) are a powerful class of generative models that exhibit remarkable empirical performance. Score-based generative modelling (SGM) consists of a ``noising'' stage, whereby a diffusion is used to gradually add Gaussian noise to data, and a generative model, which entails a ``denoising'' process defined by approximating the time-reversal of the diffusion. Existing SGMs assume that data is supported on a Euclidean space, i.e. a manifold with flat geometry. In many domains such as robotics, geoscience or protein modelling, data is often naturally described by distributions living on Riemannian manifolds and current SGM techniques are not appropriate. We introduce here Riemannian Score-based Generative Models (RSGMs), a class of generative models extending SGMs to Riemannian manifolds. We demonstrate our approach on a variety of manifolds, and in particular with earth and climate science spherical data.

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Citations in the corpus

DiffDock: Diffusion Steps, Twists, and Turns for Molecular Docking (2022, methodology): “The training and inference procedures of a score-based diffusion generative model on a Riemannian manifold consist of (1) sampling and regressing against the score of the diffusion kernel during training; and (2) sampling a geodesic random walk with the score as a drift term during inference [8].”
Protein Structure and Sequence Generation with Equivariant Denoising Diffusion Probabilistic Models (2022, methodology): “To address such limitations, recent work has extended the diffusion framework to compact Riemannian manifolds [11], which has been in turn adapted to modeling rotational diffusion as the repeated application of a heat kernel on a torus [22].”
Torsional Diffusion for Molecular Conformer Generation (2022, methodology): “To learn a generative model over this space, we apply the continuous score-based framework of Song et al. [2021], which holds with minor modiﬁcations for data distributions on compact Riemannian manifolds (such as T m ) [De Bortoli et al., 2022].”
SE(3) diffusion model with application to protein backbone generation (2023, methodology): “Proposition 2.1 (Time-reversal, De Bortoli et al. (2022)).”
Antigen-Specific Antibody Design and Optimization with Diffusion-Based Generative Models for Protein Structures (2022, methodology): “Diffusion models have also been extended to non-Euclidean data, such as data in the Riemannian manifolds [Leach et al., 2022, De Bortoli et al., 2022].”
A Survey on Generative Diffusion Models (2022, unknown): “Toaddressthis, recentmethodologiessuchasRDM[98],RGSM[97],and Boomerang[99]haveincorporateddiffusionsamplingintothe Riemannianmanifold,extendingthescoreSDEframework[11].”
Broadly applicable and accurate protein design by integrating structure prediction networks and diffusion generative models (2022, methodology): “For residue orientations, we use Brownian motion on the manifold of rotation matrices (building on refs [19,20]).”
Diffusion Models: A Comprehensive Survey of Methods and Applications (2022, background): “The Riemannian Score-Based Generative Model (RSGM) [45] accommodates a wide range of manifolds, including spheres and toruses, provided they satisfy mild conditions.”
Image Restoration with Mean-Reverting Stochastic Differential Equations (2023, background): “The general idea is to design such an SDE that gradually transforms the data distribution into fixed Gaussian noise (Song et al., 2021c; Lu et al., 2022; De Bortoli et al., 2022).”
Practical and Asymptotically Exact Conditional Sampling in Diffusion Models (2023, unknown): “Unconditional generation in Riemannian diffusion models proceeds through a geodesic random walk (De Bortoli et al., 2022).”