用于高维聚类数据的扩散模型：通过贝叶斯分类实现本征维度适应性

中文摘要 AI 辅助

扩散模型在生成建模中的经验成功推动了理论研究，包括定量误差界和表征去噪不同阶段的定性分析。我们将这两个领域结合起来，研究扩散模型对多模态高维数据结构化几何的适应性，这类数据由ℝ^D中的多个聚类组成，每个聚类具有自身的低维结构，且聚类间的分离度依赖于D。我们采用K混合高斯分布作为典型框架来捕捉这种几何结构，并建立两个理论结果。第一，我们将去噪解释为动态贝叶斯分类器：混合得分是聚类级得分的后验加权平均，且证明当信噪比达到Θ(log(KD)/D)尺度时，后验类概率以高概率集中在单个聚类上。第二，通过分别分析去噪过程的混合阶段和聚类确定阶段，我们证明即使K随D多项式增长，KL误差界也线性依赖于聚类的最大本征维度，仅差一个对数因子。这改进了环境维度的界，并将现有的低维适应性分析扩展到具有异质、近似低秩协方差的多模态分布。

英文摘要

The empirical success of diffusion models in generative modelling has motivated theoretical work, including quantitative error bounds and qualitative analyses that characterise the different phases of denoising. We bring these two areas together by studying the adaptivity of diffusion models to the structured geometry of multimodal high-dimensional data that consists of multiple clusters in $\mathbb{R}^D$, each with its own low-dimensional structure, and inter-cluster separation depending on $D$. We employ $K$-mixture Gaussian distributions as a canonical framework to capture this geometry and establish two theoretical results. First, we interpret denoising as a dynamical Bayesian classifier: the mixture score is a posterior-weighted average of cluster-wise scores, and we show that, with high probability, the posterior class probabilities concentrate on a single cluster once the signal-to-noise ratio reaches the scale $Θ(\log (KD)/D)$. Second, by separately analysing the denoising process in its mixing and cluster-commitment phases, we prove that the KL error bound depends linearly on the maximum intrinsic dimension of a cluster, up to a logarithmic factor, even when $K$ grows polynomially with $D$. This improves on ambient-dimensional bounds and extends existing low-dimensional adaptivity analyses to multimodal distributions with heterogeneous, approximately low-rank covariances.

发表机构

• Lancaster University（兰卡斯特大学）

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