On nodal domains and higher-order Cheeger inequalities of finite reversible Markov processes

Daneshgar, Amir; Javadi, Ramin; Miclo, Laurent

doi:10.1016/j.spa.2012.02.009

Cited by 18 publications

(25 citation statements)

References 12 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In [27] (see also [10]), we made the conjecture that there exists a mapping c : N Ñ R˚such that for all pS, µ, Lq as above,…”

Section: Higher Order Cheeger Inequalities In the Finite Settingmentioning

confidence: 99%

“…To avoid these problems, it is convenient to introduce the Dirichlet connectivity spectrum, which in some sense is intermediary between the usual spectrum and the connectivity spectrum. It was considered in the finite setting in [27,10] and in the continuous setting for LaplaceBeltrami operators on Euclidian or Riemannian subdomains with Dirichlet boundary conditions, for instance by Helffer, Hoffmann-Ostenhof and Terracini [15] (see also the references therein). So let us come back to a general self-adjoint Markov operator M as in the beginning of this section.…”

Section: General Higher Order Cheeger's Inequalitiesmentioning

confidence: 99%

See 1 more Smart Citation

On hyperboundedness and spectrum of Markov operators

Miclo

2014

Invent. math.

Self Cite

View full text Add to dashboard Cite

Consider an ergodic Markov operator M reversible with respect to a probability measure µ on a general measurable space. It is shown that if M is bounded from L 2 pµq to L p pµq, where p ą 2, then it admits a spectral gap. This result answers positively a conjecture raised by Høegh-Krohn and Simon [31] in a semi-group context. The proof is based on isoperimetric considerations and especially on Cheeger inequalities of higher order for weighted finite graphs recently obtained by Lee, Gharan and Trevisan [23]. It provides a quantitative link between hyperboundedness and an eigenvalue different from the spectral gap in general. In addition, the usual Cheeger inequality is extended to the higher eigenvalues in the compact Riemannian setting.

show abstract

“…In [27] (see also [10]), we made the conjecture that there exists a mapping c : N Ñ R˚such that for all pS, µ, Lq as above,…”

Section: Higher Order Cheeger Inequalities In the Finite Settingmentioning

confidence: 99%

Section: General Higher Order Cheeger's Inequalitiesmentioning

confidence: 99%

On hyperboundedness and spectrum of Markov operators

Miclo

2014

Invent. math.

Self Cite

View full text Add to dashboard Cite

show abstract

“…Recently, theoretical justifications of some of these algorithms have been given. For example, [LOT12,LRTV12,DJM12] used spectral embedding to prove higher order variants of Cheeger's inequality, namely, that a graph can be partitioned into k subsets each defining a sparse cut if and only if the kth smallest eigenvalue of the normalized Laplacian is close to zero.…”

Section: Introductionmentioning

confidence: 99%

Sharp Bounds on Random Walk Eigenvalues via Spectral Embedding

Lyons

Gharan

2017

International Mathematics Research Notices

View full text Add to dashboard Cite

Spectral embedding of graphs uses the top k non-trivial eigenvectors of the random walk matrix to embed the graph into R k . The primary use of this embedding has been for practical spectral clustering algorithms [SM00, NJW01]. Recently, spectral embedding was studied from a theoretical perspective to prove higher order variants of Cheeger's inequality [LOT12,LRTV12].We use spectral embedding to provide a unifying framework for bounding all the eigenvalues of graphs. For example, we show that for any finite connected graph with n vertices and all k ≥ 2, the kth largest eigenvalue is at most 1 − Ω(k 3 /n 3 ), which extends the only other such result known, which is for k = 2 only and is due to [LO81]. This upper bound improves to 1 − Ω(k 2 /n 2 ) if the graph is regular. We generalize these results, and we provide sharp bounds on the spectral measure of various classes of graphs, including vertex-transitive graphs and infinite graphs, in terms of specific graph parameters like the volume growth.As a consequence, using the entire spectrum, we provide (improved) upper bounds on the return probabilities and mixing time of random walks with considerably shorter and more direct proofs. Our work introduces spectral embedding as a new tool in analyzing reversible Markov chains. Furthermore, building on [Lyo05], we design a local algorithm to approximate the number of spanning trees of massive graphs.

show abstract

“…The Cheeger estimate was first generalized to graphs by Dodziuk [Dod84] and Alon-Milman [AM85] independently. Miclo introduced higher order Cheeger constants and conjectured related higher order Cheeger estimates, see [Mic08,DJM12]. The conjecture was proved by Lee, Oveis Gharan and Tevisan [LOGT14] via random partition methods on graphs.…”

Section: Introductionmentioning

confidence: 98%