arXiv:2605.08777stat.MLcs.LG2026-05

用扩散过程量化高维分布的模式分离,捕捉传统方法忽略的结构信息。

Measuring and Decomposing Mode Separation via the Canonical Diffusion

论文配图:Measuring and Decomposing Mode Separation via the Canonical Diffusion
图 1 · 摘自论文原文
  • 基于密度内在的可逆扩散过程,提取对势垒敏感的统计量。
  • 提出SSA和DA两个读数,能识别低方差但高亚稳态的主方向。
  • 适用于生成模型、分子动力学等场景,仅需采样和得分函数。

模式分离指分布以势垒分隔的聚类程度,是高维密度的基本几何特性,却难以量化。现有工具不足:微分熵随扩散上升,不关心碎片化;主成分分析按方差排序,无视势垒;互信息需混合分解,通常不可得。本文通过密度内在的唯一可逆扩散过程(以f为平稳分布,常数扩散系数)测量模式分离。从自协方差矩阵中提取两个读数:SSA(平方自相关之和),为标量屏障敏感度量;DA(主导自相关方向),按亚稳性而非方差排序的线性投影。在各向同性高斯零假设下,推导出经验自协方差的闭式谱,推广了Marchenko-Pastur理论,解析上界确定了提取DA的时间延迟。两者仅需样本和得分函数,可通过Tweedie恒等式利用预训练得分生成模型扩展至高维。应用于三类场景:(i) 合成高斯混合,SSA追踪互信息;(ii) SDXL文生图,SSA与DA捕获熵和PCA遗漏的结构;(iii) 丙氨酸二肽分子动力学,仅用静态样本,DA恢复已知慢骨架二面角。

原文摘要 · Abstract (English)

Mode separation, namely how sharply a distribution fragments into barrier-separated clusters, is a fundamental geometric property of densities, difficult to quantify in high dimensions. It is structurally distinct from dispersion, yet existing tools fall short: differential entropy rises with spread regardless of fragmentation, PCA orders directions by variance regardless of barriers, and mutual information requires a mixture decomposition one usually does not have. We measure mode separation through a single stochastic process intrinsic to the density: a unique reversible diffusion with $f$ as its stationary distribution and constant scalar diffusion coefficient. We extract two readouts from its autocovariance matrix: SSA (Sum of Squared Autocorrelations), a scalar barrier-sensitive measure; and DA (Dominant Autocorrelation directions), linear projections ordered by metastability rather than variance. Under an isotropic-Gaussian null, we derive a closed-form spectrum for the empirical autocovariance that generalizes Marchenko--Pastur, with an analytic upper edge that selects the lag at which DA is read off. Both readouts use only samples and a score function, scaling to high dimensions through pretrained score-based generative models via Tweedie's identity. We apply our framework to three settings: (i) synthetic Gaussian mixtures, where SSA tracks mutual information; (ii) SDXL text-to-image generations, where SSA and DA capture structure that entropy and PCA miss; and (iii) molecular dynamics of alanine dipeptide, where DA recovers the known slow backbone dihedrals from static samples alone.

扩散模型模式分离生成模型高维分析

Thank you to arXiv for use of its open access interoperability. PaperDance 不是 arXiv 官方产品;中文卡片由大模型生成,请以原文为准。