arXiv:2602.04404cs.LGcond-mat.dis-nn2026-02被引 10

揭示扩散模型生成过程中的物种分化机制,适用于任意可分类数据分布。

Theory of Speciation Transitions in Diffusion Models with General Class Structure

  • 基于贝叶斯分类定义类别结构,用自由熵差刻画分化时间。
  • 理论扩展至非高斯、高阶特征差异的分布,预测多阶段分化现象。
  • 在伊辛模型与异协方差高斯混合上验证,可解析求解分化时间。

扩散模型通过逆转随机扩散过程生成数据,逐步将噪声转化为目标分布的结构化样本。近期理论发现,该反向动态可能经历剧烈的定性转变,即物种分化(speciation)现象,此时轨迹被动态锁定于特定数据类别。现有分析局限于可通过一阶矩区分的类别,如均值分离的高斯混合模型。本文发展了一般化物种分化理论,适用于任意具有明确类别的目标分布。我们通过贝叶斯分类形式化类别结构,并以类间自由熵差刻画分化时间。该准则恢复了已有高斯混合模型结果,同时拓展至无法通过一阶矩区分、依赖高阶或集体特征的场景。框架支持多类别,并预测存在逐级细化的分化时间序列。我们在两个解析可解案例中验证:不同温度下的一维伊辛模型混合,以及具有不同协方差结构的零均值高斯混合。在伊辛情形中,通过映射到随机场伊辛模型并使用复制法,获得分化时间的显式表达式。结果为扩散生成模型中的物种分化提供了统一且广泛适用的描述。

原文摘要 · Abstract (English)

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.

扩散模型生成模型理论分析物种分化

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