# A Theory of Speciation in Generative Diffusion Models on Compact Riemannian Manifolds

arXiv: https://arxiv.org/abs/2608.23798 · HTML: https://mlsanity.com/paper/2608.23798/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models.
- Builds on 23 of its 53 references that are in the map.
- First posted on arXiv 24 Aug 2026.

## Placed in

- Diffusion models / [Diffusion foundations and sampling](https://mlsanity.com/diffusion/diffusion-foundations-and-sampling/)

## Builds on (most cited first, 23 of 23)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Generative Modeling by Estimating Gradients of the Data Distribution](https://mlsanity.com/paper/1907.05600/) (2019)
- [Deep Learning Face Attributes in the Wild](https://mlsanity.com/paper/1411.7766/) (2014)
- [The Intrinsic Dimension of Images and Its Impact on Learning](https://mlsanity.com/paper/2104.08894/) (2021)
- [Riemannian Score-Based Generative Modelling](https://mlsanity.com/paper/2202.02763/) (2022)
- [Dynamical Regimes of Diffusion Models](https://mlsanity.com/paper/2402.18491/) (2024)
- [Understanding the Latent Space of Diffusion Models through the Lens of Riemannian Geometry](https://mlsanity.com/paper/2307.12868/) (2023)
- [Riemannian Diffusion Models](https://mlsanity.com/paper/2208.07949/) (2022)
- [Spontaneous Symmetry Breaking in Generative Diffusion Models](https://mlsanity.com/paper/2305.19693/) (2023)
- [The statistical thermodynamics of generative diffusion models: Phase transitions, symmetry breaking and critical instability](https://mlsanity.com/paper/2310.17467/) (2023)
- [Manifolds, Random Matrices and Spectral Gaps: The geometric phases of generative diffusion](https://mlsanity.com/paper/2410.05898/) (2024)
- [Pulling back information geometry](https://mlsanity.com/paper/2106.05367/) (2021)
- [Large-scale Log-determinant Computation through Stochastic Chebyshev Expansions](https://mlsanity.com/paper/1503.06394/) (2015)
- [Manifold Learning by Mixture Models of VAEs for Inverse Problems](https://mlsanity.com/paper/2303.15244/) (2023)
- [The Information Dynamics of Generative Diffusion](https://mlsanity.com/paper/2508.19897/) (2025)
- [Closing the ODE-SDE gap in score-based diffusion models through the Fokker-Planck equation](https://mlsanity.com/paper/2311.15996/) (2023)
- [Counterfactual Explanations via Riemannian Latent Space Traversal](https://mlsanity.com/paper/2411.02259/) (2024)
- [How Out-of-Equilibrium Phase Transitions can Seed Pattern Formation in Trained Diffusion Models](https://mlsanity.com/paper/2603.20092/) (2026)
- [Connecting Neural Models Latent Geometries with Relative Geodesic Representations](https://mlsanity.com/paper/2506.01599/) (2025)
- [PCAE: Learning Ordered Representations in Latent Space for Intrinsic Dimension Estimation via Principal Component Autoencoder](https://mlsanity.com/paper/2601.19179/) (2026)
- [Riemannian simplices and triangulations](https://mlsanity.com/paper/1406.3740/) (2014)
- [Estimating Dataset Dimension via Singular Metrics under the Manifold Hypothesis: Application to Inverse Problems](https://mlsanity.com/paper/2507.07291/) (2025)

## Built on by (most cited first, 1 of 1)

- [When Does a Diffusion Model Decide What to Draw ?](https://mlsanity.com/paper/2610.05645/) (2026)

## Abstract (arXiv)

Speciation in generative diffusion models denotes the emergence of distinct stable branches during denoising, through which initially undifferentiated trajectories progressively commit to different data classes. In this work we develop an intrinsic theory of speciation for diffusion models supported on compact Riemannian manifolds: the aim is to go beyond existing theoretical descriptions, which usually identify speciation with a symmetric pitchfork bifurcation and assume to work in a large-dimensional space. We characterize speciation by bifurcations of the critical points of the evolving probability density. A spectral heat-kernel representation makes explicit the role of the manifold geometry, while Poincar'e-Hopf and Morse theory impose global constraints on the number and type of score equilibria and reveal topologically-imposed geometrical modes. For mixtures of heat kernels, we prove that generic speciation events have a one-dimensional critical kernel and admit an A2 fold normal form; pitchforks and simultaneous multidirectional transitions arise from nongeneric symmetric configurations. We derive geometry-dependent estimates of speciation times for bimodal mixtures and Riemannian regular simplices. We further establish structural stability of nondegenerate folds under score perturbations and show that the first-order time shift is determined solely by the component of the score error along the critical direction. The theory is illustrated on the sphere using mixtures of von Mises-Fisher distributions, where pitchfork and saddle-node bifurcations, topological modes, and hierarchical multiple speciations are observed. Finally, a chart-based intrinsic score-learning scheme based on neural networks contrasts the theoretically predicted transitions on prototypal and more complex datasets.