Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Antonio Corbo Esposito, Gianpaolo Piscitelli

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.

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

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 -Laplacian. In this paper, we firstly give new notations to describe the paths, among critical eigenvectors of the graph -Laplacian, realizing sets with prescribed genus. We introduce the pseudo-orthogonality to characterize , a special eigenvalue for the graph -Laplacian. Furthermore, we use it to give an upper bound for the third graph Cheeger constant , that is . This is a first step for proving that the -th Cheeger constant is the minimum of the -Laplacian Raylegh quotient among vectors that are pseudo-orthogonal to the vectors realizing the previous Cheeger constants. Eventually, we apply these results to give a method and a numerical algorithm to compute , based on a generalized inverse power method.

Paper Structure

This paper contains 16 sections, 18 theorems, 124 equations.

Table of Contents

  1. Introduction
  2. The Graph 1-Laplacian Eigenvalue Problem
  3. The first eigenpair
  4. The role of the nodal domains
  5. The first and the second Cheeger constants
  6. Paths among Eigenvalues
  7. New notations to treat 1-Laplacian
  8. The behaviour of the eigenvectors
  9. Paths among eigenvalues
  10. The Pseudo-orthogonality for characterizing the Cheeger Constants
  11. The asymptotic behaviour of the graph p-Laplacian eigenvalue in the continuous case
  12. The role of the orthogonality
  13. The graph Cheeger constants in form of continuous optimizations
  14. The pseudo-orthogonality
  15. Proof of the main results
  16. ...and 1 more sections

Key Result

Theorem 1.1

Let $G=(V,E)$ be a graph, then

Theorems & Definitions (38)

  • Theorem 1.1
  • Proposition 2.1: Cor. 2.5 Cha
  • Proposition 2.2
  • Proposition 2.3
  • Remark 2.4
  • Proposition 2.5
  • Proposition 2.6
  • proof
  • Proposition 2.7
  • Proposition 3.1
  • ...and 28 more