Abstract

We introduce a set of multi-way dual Cheeger constants and prove universal higher-order dual Cheeger inequalities for eigenvalues of normalized Laplace operators on weighted finite graphs. Our proof proposes a new spectral clustering phenomenon deduced from metrics on real projective spaces. We further extend those results to a general reversible Markov operator and find applications in characterizing its essential spectrum.

Highlights

  • Introduction and main ideasCheeger constant, encoding global connectivity properties of the underlying space, was invented by Cheeger [7] and related to the first non-zero eigenvalue of the Laplace– Beltrami operator on a compact Riemannian manifold, which is well-known as Cheeger inequality

  • We introduce a set of multi-way dual Cheeger constants and prove universal higher-order dual Cheeger inequalities for eigenvalues of normalized Laplace operators on weighted finite graphs

  • Our proof proposes a new spectral clustering phenomenon deduced from metrics on real projective spaces. We further extend those results to a general reversible Markov operator and find applications in characterizing its essential spectrum

Read more

Summary

Introduction and main ideas

Cheeger constant, encoding global connectivity properties of the underlying space, was invented by Cheeger [7] and related to the first non-zero eigenvalue of the Laplace– Beltrami operator on a compact Riemannian manifold, which is well-known as Cheeger inequality. In discrete setting and conjectured related higher-order Cheeger inequalities universal for any weighted graph This conjecture was solved by Lee, Oveis Gharan and Trevisan [26] by bringing in the powerful tool of random metric partitions developed originally in theoretical computer science. We prove higher-order dual Cheeger inequalities, i.e., we derive estimates for the spectral gaps 2 − λN−k+1 in terms of h(k), which hold universally for any weighted finite graph (see Theorem 1.2). This completes the picture about graph spectra and (dual) isoperimetric constants. A further discussion about the relations between hP (k) and hP (k) enables us to arrive at sup hP (k) > 0 ⇔ −1 < λess(P ) ≤ λess(P ) < 1

Statements of main results
Clustering on real projective spaces
Organization of the paper
Spectral theory for normalized graph Laplacian
Padded random partitions of doubling metric space
The metric for clustering via top k eigenfunctions
Real projective space with a rough metric
Spreading lemma
Localization lemma
Trees and cycles
Findings
Essential spectrum of reversible Markov operators
Full Text
Published version (Free)

Talk to us

Join us for a 30 min session where you can share your feedback and ask us any queries you have

Schedule a call