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
Summary
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
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