The Pseudo-orthogonality for Graph $1$-Laplacian Eigenvectors and Applications to Higher Cheeger Constants and Data Clustering

TL;DR

This paper introduces pseudo-orthogonality for graph 1-Laplacian eigenvectors, providing new bounds for higher Cheeger constants and developing an algorithm for their computation, advancing spectral clustering methods.

Contribution

It proposes the concept of pseudo-orthogonality to characterize eigenvalues related to higher Cheeger constants and offers a numerical method to compute these constants.

Findings

Established an upper bound for the third Cheeger constant using $m_3(G)$.

Introduced a generalized inverse power method for computing $m_3(G)$.

Connected the $k$-th Cheeger constant to eigenvalues and pseudo-orthogonality.

Abstract

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.