图1-拉普拉斯算子特征向量的伪正交性及其在高阶Cheeger常数和数据聚类中的应用
The Pseudo-orthogonality for Graph $1$-Laplacian Eigenvectors and Applications to Higher Cheeger Constants and Data Clustering
AI总结：
该研究引入伪正交性刻画图1-拉普拉斯的特殊特征值m₃(G)，给出第三个Cheeger常数的上界，并基于广义逆幂法提出计算m₃(G)的算法，为谱聚类相关理论提供支撑。
摘要：
数据聚类问题旨在将数据集划分为指定数目的同构数据组，这是一个NP难问题，可在谱图理论中得到松弛，该理论中图的最优割与图1-拉普拉斯算子的特征值相关。本文首先提出新的记号，用于描述图1-拉普拉斯算子的临界特征向量中实现指定亏格集合的路径。我们引入伪正交性来刻画m₃(G)——图1-拉普拉斯算子的一个特殊特征值。进一步地，我们利用该性质给出第三个图Cheeger常数h₃(G)的上界，即h₃(G) ≤ m₃(G)。这是证明第k个Cheeger常数是与前k-1个Cheeger常数的实现向量伪正交的向量中1-拉普拉斯瑞利商最小值的第一步。最后，我们应用这些结果提出了一种基于广义逆幂法计算m₃(G)的方法和数值算法。
英文摘要：
The data clustering problem consists in dividing a data set into prescribed groups of homogeneous data. This is a NP-hard problem that can be relaxed in the spectral graph theory, where the optimal cuts of a graph are related to the eigenvalues of graph $1$-Laplacian. In this paper, we firstly give new notations to describe the paths, among critical eigenvectors of the graph $1$-Laplacian, realizing sets with prescribed genus. We introduce the pseudo-orthogonality to characterize $m_3(G)$, a special eigenvalue for the graph $1$-Laplacian. Furthermore, we use it to give an upper bound for the third graph Cheeger constant $h_3(G)$, that is $h_3(G) \le m_3(G)$. This is a first step for proving that the $k$-th Cheeger constant is the minimum of the $1$-Laplacian Raylegh quotient among vectors that are pseudo-orthogonal to the vectors realizing the previous $k-1$ Cheeger constants. Eventually, we apply these results to give a method and a numerical algorithm to compute $m_3(G)$, based on a generalized inverse power method.
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