Multi-way dual Cheeger constants and spectral bounds of graphs

Liu, Shiping

doi:10.1016/j.aim.2014.09.023

Cited by 26 publications

(42 citation statements)

References 37 publications

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“…In particular, every cycle graph C N with N vertices satisfies CD(0, ∞). Moreover, we have for the graph C N (see, e.g., [29,Proposition 7.4]), 14) which is in line with the Cheeger inequality and Buser inequality, and also…”

Section: Motivation and A Dual Buser Inequalitysupporting

confidence: 73%

Curvature and Higher Order Buser Inequalities for the Graph Connection Laplacian

Liu¹,

Münch²,

Peyerimhoff³

2019

SIAM J. Discrete Math.

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We study the eigenvalues of the connection Laplacian on a graph with an orthogonal group or unitary group signature. We establish higher order Buser type inequalities, i.e., we provide upper bounds for eigenvalues in terms of Cheeger constants in the case of nonnegative Ricci curvature. In this process, we discuss the concepts of Cheeger type constants and a discrete Ricci curvature for connection Laplacians and study their properties systematically. The Cheeger constants are defined as mixtures of the expansion rate of the underlying graph and the frustration index of the signature. The discrete curvature, which can be computed efficiently via solving semidefinite programming problems, has a characterization by the heat semigroup for functions combined with a heat semigroup for vector fields on the graph.

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Section: Motivation and A Dual Buser Inequalitysupporting

confidence: 73%

Curvature and Higher Order Buser Inequalities for the Graph Connection Laplacian

Liu¹,

Münch²,

Peyerimhoff³

2019

SIAM J. Discrete Math.

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“…In [38], it is argued that the largest eigenvalue of a normalized Laplacian can be used to bound from below and above the bipartiteness ratio, which measures the extent to which the graph is approximated by a bipartite graph. Its higher order version is also studied [26]. It would be interesting to generalize these extended Cheeger inequalities for submodular transformations.…”

Section: Discussionmentioning

confidence: 99%

Cheeger Inequalities for Submodular Transformations

Yoshida¹

2019

Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms

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The Cheeger inequality for undirected graphs, which relates the conductance of an undirected graph and the second smallest eigenvalue of its normalized Laplacian, is a cornerstone of spectral graph theory. The Cheeger inequality has been extended to directed graphs and hypergraphs using normalized Laplacians for those, that are no longer linear but piecewise linear transformations.In this paper, we introduce the notion of a submodular transformation F : {0, 1} n → R m , which applies m submodular functions to the n-dimensional input vector, and then introduce the notions of its Laplacian and normalized Laplacian. With these notions, we unify and generalize the existing Cheeger inequalities by showing a Cheeger inequality for submodular transformations, which relates the conductance of a submodular transformation and the smallest non-trivial eigenvalue of its normalized Laplacian. This result recovers the Cheeger inequalities for undirected graphs, directed graphs, and hypergraphs, and derives novel Cheeger inequalities for mutual information and directed information.Computing the smallest non-trivial eigenvalue of a normalized Laplacian of a submodular transformation is NP-hard under the small set expansion hypothesis. In this paper, we present a polynomial-time O(log n)-approximation algorithm for the symmetric case, which is tight, and a polynomial-time O(log 2 n + log n · log m)-approximation algorithm for the general case.We expect the algebra concerned with submodular transformations, or submodular algebra, to be useful in the future not only for generalizing spectral graph theory but also for analyzing other problems that involve piecewise linear transformations, e.g., deep learning.

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“…Thus, each eigenvalue λ i (∆ s µ ) of ∆ s µ is an eigenvalue of L s with doubled multiplicity. If we denote the Euclidean norm in R l by · , Bandeira, Singer and Spielman define a (partial) 1 frustration constant as 31) and prove that…”

Section: Cheeger's Inequalitymentioning

confidence: 99%

“…To prove higher order Cheeger inequalities, we develop new multi-way spectral clustering algorithms using metrics on lens spaces and complex projective spaces. This provides a deeper understanding of earlier spectral clustering algorithms via metrics on real projective spaces presented in [31] and [3]. These clustering algorithms were initially designed to find almost bipartite subgraphs of a given graph, [31], and then extended to find almost balanced subgraphs of a signed graph, [3].…”

Section: Introductionmentioning

confidence: 99%

“…These clustering algorithms were initially designed to find almost bipartite subgraphs of a given graph, [31], and then extended to find almost balanced subgraphs of a signed graph, [3]. While all operators studied in [31,3] are bounded, we show that finding proper metrics for clustering is also useful for unbounded operators: the spectral clustering algorithms via metrics on complex projective spaces are crucial to prove the higher order Cheeger inequalities of the magnetic Laplacian on a closed Riemannian manifold (Lemma 7.8).…”

Section: Introductionmentioning

confidence: 99%

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Frustration index and Cheeger inequalities for discrete and continuous magnetic Laplacians

et al. 2015

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We discuss a Cheeger constant as a mixture of the frustration index and the expansion rate, and prove the related Cheeger inequalities and higher order Cheeger inequalities for graph Laplacians with cyclic signatures, discrete magnetic Laplacians on finite graphs and magnetic Laplacians on closed Riemannian manifolds. In this process, we develop spectral clustering algorithms for partially oriented graphs and multi-way spectral clustering algorithms via metrics in lens spaces and complex projective spaces. As a byproduct, we give a unified viewpoint of Harary's structural balance theory of signed graphs and the gauge invariance of magnetic potentials.

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Multi-way dual Cheeger constants and spectral bounds of graphs

Cited by 26 publications

References 37 publications

Curvature and Higher Order Buser Inequalities for the Graph Connection Laplacian

Curvature and Higher Order Buser Inequalities for the Graph Connection Laplacian

Cheeger Inequalities for Submodular Transformations

Frustration index and Cheeger inequalities for discrete and continuous magnetic Laplacians

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