发表机构
University of Milan（米兰大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本研究建立紧致黎曼流形上生成式扩散模型的物种形成内蕴理论，刻画其分岔机制，推导相关估计与稳定性结论，经球面实验及神经网络学习方案验证。
AI 中文摘要
生成式扩散模型中的物种形成指去噪过程中出现不同稳定分支，初始未分化轨迹通过这些分支逐步归属不同数据类。本工作针对支撑在紧致黎曼流形上的扩散模型，建立了物种形成的内蕴理论，旨在突破现有理论描述——现有理论通常将物种形成等同于对称叉形分岔，并假设在高维空间中工作。我们通过演化概率密度临界点的分岔来刻画物种形成；谱热核表示明确了流形几何的作用，而庞加莱-霍普夫定理与莫尔斯理论对得分平衡点的数量和类型施加全局约束，并揭示拓扑强制的几何模式。对于热核混合模型，我们证明一般物种形成事件具有一维临界核，且满足A2折正则型；叉形分岔与同时多向转变源于非一般对称构型。我们推导了双峰混合模型与黎曼正则单纯形的几何依赖物种形成时间估计，进一步建立非退化折在得分扰动下的结构稳定性，表明一阶时间偏移仅由得分误差沿临界方向的分量决定。该理论在球面上用冯·米塞斯-费希尔分布混合模型验证，观察到叉形分岔、鞍结分岔、拓扑模式及分层多重物种形成；最后，基于神经网络的内蕴得分学习方案（依托图表），在原型及更复杂数据集上验证了理论预测的转变。
英文摘要
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 Poincaré-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.
Comments48 pages, 15 figures
