Diffusion Models generate data by reversing a stochastic diffusion process, progressively transforming noise into structured samples drawn from a target distribution. Recent theoretical work has shown that this backward dynamics can undergo sharp qualitative transitions, known as speciation transitions, during which trajectories become dynamically committed to data classes. Existing theoretical analyses, however, are limited to settings where classes are identifiable through first moments, such as mixtures of Gaussians with well-separated means. In this work, we develop a general theory of speciation in diffusion models that applies to arbitrary target distributions admitting well-defined classes. We formalize the notion of class structure through Bayes classification and characterize speciation times in terms of free-entropy difference between classes. This criterion recovers known results in previously studied Gaussian-mixture models, while extending to situations in which classes are not distinguishable by first moments and may instead differ through higher-order or collective features. Our framework also accommodates multiple classes and predicts the existence of successive speciation times associated with increasingly fine-grained class commitment. We illustrate the theory on two analytically tractable examples: mixtures of one-dimensional Ising models at different temperatures and mixtures of zero-mean Gaussians with distinct covariance structures. In the Ising case, we obtain explicit expressions for speciation times by mapping the problem onto a random-field Ising model and solving it via the replica method. Our results provide a unified and broadly applicable description of speciation transitions in diffusion-based generative models.

翻译：扩散模型通过逆转随机扩散过程生成数据，逐步将噪声转化为从目标分布中抽取的结构化样本。近期理论研究证明，这种反向动力学可能经历急剧的定性转变（称为物种形成转变），在此过程中轨迹会动态地固定于数据类别。然而，现有理论分析仅限于类别可通过一阶矩识别的场景，例如均值分离良好的高斯混合分布。本研究建立了适用于任意具有明确定义类别的目标分布的扩散模型物种形成一般理论。我们通过贝叶斯分类形式化类别结构的概念，并依据类别间的自由熵差来刻画物种形成时间。该准则恢复了先前研究的高斯混合模型中已知结果，同时扩展至类别无法通过一阶矩区分、而可能通过高阶或集体特征区分的场景。我们的框架还兼容多类别情况，并预测存在与日益细粒度类别固定相关联的连续物种形成时间。我们通过两个可解析处理的示例阐明该理论：不同温度下的一维伊辛模型混合分布，以及具有不同协方差结构的零均值高斯混合分布。在伊辛模型案例中，通过将问题映射到随机场伊辛模型并采用复本方法求解，我们获得了物种形成时间的显式表达式。本研究结果为基于扩散的生成模型中物种形成转变提供了统一且广泛适用的理论描述。