扩散映射、谱聚类与Fokker-Planck算子的特征函数
Diffusion Maps, Spectral Clustering and Eigenfunctions of Fokker-Planck operators
AI总结：
本文提出基于扩散过程的谱聚类和降维算法的概率解释，通过扩散距离和Fokker-Planck算子特征函数，为使用归一化图拉普拉斯特征向量的方法提供数学依据。
摘要：
本文提出了一种基于扩散的谱聚类和降维算法的概率解释，这些算法使用归一化图拉普拉斯算子的特征向量。给定所有点的成对邻接矩阵，我们定义了任意两个数据点之间的扩散距离，并表明在一定的均方误差准则下，由相应马尔可夫矩阵的前几个特征向量给出的数据的低维表示是最优的。此外，假设数据点是从密度 $p(\x) = e^{-U(\x)}$ 中随机抽取的样本，我们将这些特征向量识别为在势 $2U(\x)$ 中具有反射边界条件的Fokker-Planck算子的特征函数的离散近似。最后，应用关于连续Fokker-Planck算子的特征值和特征函数的已知结果，我们为基于这些前几个特征向量的谱聚类和降维算法的成功提供了数学依据。该分析从扩散过程的特征角度阐明了许多关于谱聚类算法的经验发现。
英文摘要：
This paper presents a diffusion based probabilistic interpretation of spectral clustering and dimensionality reduction algorithms that use the eigenvectors of the normalized graph Laplacian. Given the pairwise adjacency matrix of all points, we define a diffusion distance between any two data points and show that the low dimensional representation of the data by the first few eigenvectors of the corresponding Markov matrix is optimal under a certain mean squared error criterion. Furthermore, assuming that data points are random samples from a density $p(\x) = e^{-U(\x)}$ we identify these eigenvectors as discrete approximations of eigenfunctions of a Fokker-Planck operator in a potential $2U(\x)$ with reflecting boundary conditions. Finally, applying known results regarding the eigenvalues and eigenfunctions of the continuous Fokker-Planck operator, we provide a mathematical justification for the success of spectral clustering and dimensional reduction algorithms based on these first few eigenvectors. This analysis elucidates, in terms of the characteristics of diffusion processes, many empirical findings regarding spectral clustering algorithms.
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