Cheeger Inequalities Research Articles

Cheeger inequalities for graph limits

Cheeger inequalities for graph limits

Multiway Spectral Graph Partitioning: Cut Functions, Cheeger Inequalities, and a Simple Algorithm

Multiway Spectral Graph Partitioning: Cut Functions, Cheeger Inequalities, and a Simple Algorithm

Hypergraphs with edge-dependent vertex weights: p-Laplacians and spectral clustering.

We study p-Laplacians and spectral clustering for a recently proposed hypergraph model that incorporates edge-dependent vertex weights (EDVW). These weights can reflect different importance of vertices within a hyperedge, thus conferring the hypergraph model higher expressivity and flexibility. By constructing submodular EDVW-based splitting functions, we convert hypergraphs with EDVW into submodular hypergraphs for which the spectral theory is better developed. In this way, existing concepts and theorems such as p-Laplacians and Cheeger inequalities proposed under the submodular hypergraph setting can be directly extended to hypergraphs with EDVW. For submodular hypergraphs with EDVW-based splitting functions, we propose an efficient algorithm to compute the eigenvector associated with the second smallest eigenvalue of the hypergraph 1-Laplacian. We then utilize this eigenvector to cluster the vertices, achieving higher clustering accuracy than traditional spectral clustering based on the 2-Laplacian. More broadly, the proposed algorithm works for all submodular hypergraphs that are graph reducible. Numerical experiments using real-world data demonstrate the effectiveness of combining spectral clustering based on the 1-Laplacian and EDVW.

On a class of Cheeger inequalities

We study a general version of the Cheeger inequality by considering the shape functional mathcal {F}_{p,q}(Omega )=lambda _p^{1/p}(Omega )/lambda _q^{1/q}(Omega ). The infimum and the supremum of mathcal {F}_{p,q} are studied in the class of all domains Omega of mathbb {R}^d and in the subclass of convex domains. In the latter case the issue concerning the existence of an optimal domain for mathcal {F}_{p,q} is discussed.

Cheeger Inequalities for the Discrete Magnetic Laplacian

Cheeger Inequalities for the Discrete Magnetic Laplacian

A Cheeger Cut for Uniform Hypergraphs

The graph Cheeger constant and Cheeger inequalities are generalized to the case of hypergraphs whose edges have the same cardinality. In particular, it is shown that the second largest eigenvalue of the generalized normalized Laplacian is bounded both above and below by the generalized Cheeger constant, and the corresponding eigenfunctions can be used to approximate the Cheeger cut.

The equality case in Cheeger's and Buser's inequalities on [formula omitted] spaces

We prove that the sharp Buser's inequality obtained in the framework of RCD(1,∞) spaces by the first two authors [29] is rigid, i.e. equality is obtained if and only if the space splits isomorphically a Gaussian. The result is new even in the smooth setting. We also show that the equality in Cheeger's inequality is never attained in the setting of RCD(K,∞) spaces with finite diameter or positive curvature, and we provide several examples of spaces with Ricci curvature bounded below where these assumptions are not satisfied and the equality is attained.

Spectral sparsification of simplicial complexes for clustering and label propagation

As a generalization of the use of graphs to describe pairwise interactions, simplicial complexes can be used to model higher-order interactions between three or more objects in complex systems. There has been a recent surge in activity for the development of data analysis methods applicable to simplicial complexes, including techniques based on computational topology, higher-order random processes, generalized Cheeger inequalities, isoperimetric inequalities, and spectral methods. In particular, spectral learning methods (e.g. label propagation and clustering) that directly operate on simplicial complexes represent a new direction emerging from the confluence of computational topology and machine learning. To apply spectral methods in learning to massive datasets modeled as simplicial complexes, we sparsify simplicial complexes, while preserving the spectrum of the associated Laplacian operators. We show that the theory of Spielman and Srivastava for the sparsification of graphs extends to simplicial complexes via the up Laplacian. In particular, we introduce a generalized effective resistance for simplexes, provide an algorithm for sparsifying simplicial complexes at a fixed dimension, and give a specific version of the generalized Cheeger inequality for weighted simplicial complexes. Finally, we introduce higher-order generalizations of spectral clustering and label propagation for simplicial complexes and demonstrate via experiments the utility of the proposed spectral sparsification method for these applications.

Cheeger constants, structural balance, and spectral clustering analysis for signed graphs

We introduce a family of multi-way Cheeger-type constants {hkσ,k=1,2,…,n} on a signed graph Γ=(G,σ) such that hkσ=0 if and only if Γ has k balanced connected components. These constants are switching invariant and bring together in a unified viewpoint a number of important graph-theoretical concepts, including the classical Cheeger constant, those measures of bipartiteness introduced by Desai-Rao, Trevisan, Bauer–Jost, respectively, on unsigned graphs, and the frustration index (originally called the line index of balance by Harary) on signed graphs. We further unify the (higher-order or improved) Cheeger and dual Cheeger inequalities for unsigned graphs as well as the underlying algorithmic proof techniques by establishing their corresponding versions on signed graphs. In particular, we develop a spectral clustering method for finding k almost-balanced subgraphs, each defining a sparse cut. The proper metric for such a clustering is the metric on a real projective space. We also prove estimates of the extremal eigenvalues of signed Laplace matrix in terms of number of signed triangles (3-cycles).

General Cheeger inequalities for [formula omitted]-Laplacians on graphs

We prove Cheeger inequalities for p-Laplacians on finite and infinite weighted graphs. Unlike in previous works, we do not impose boundedness of the vertex degree, nor do we restrict ourselves to the normalized Laplacian and, more generally, we do not impose any boundedness assumption on the geometry. This is achieved by a novel definition of the measure of the boundary which uses the idea of intrinsic metrics. For the non-normalized case, our bounds on the spectral gap of p-Laplacians are already significantly better for finite graphs and for infinite graphs they yield non-trivial bounds even in the case of unbounded vertex degree. We, furthermore, give upper bounds by the Cheeger constant and by the exponential volume growth of distance balls.

Cheeger inequalities for absorbing Markov chains

Abstract We construct Cheeger-type bounds for the second eigenvalue of a substochastic transition probability matrix in terms of the Markov chain's conductance and metastability (and vice versa) with respect to its quasistationary distribution, extending classical results for stochastic transition matrices.

Almost Optimal Local Graph Clustering Using Evolving Sets

Spectral partitioning is a simple, nearly linear time algorithm to find sparse cuts, and the Cheeger inequalities provide a worst-case guarantee for the quality of the approximation found by the algorithm. A local graph partitioning algorithm finds a set of vertices with small conductance (i.e., a sparse cut) by adaptively exploring part of a large graph G , starting from a specified vertex. For the algorithm to be local, its complexity must be bounded in terms of the size of the set that it outputs, with at most a weak dependence on the number n of vertices in G . Previous local partitioning algorithms find sparse cuts using random walks and personalized PageRank [Spielman and Teng 2013; Andersen et al. 2006]. In this article, we introduce a simple randomized local partitioning algorithm that finds a sparse cut by simulating the volume-biased evolving set process , which is a Markov chain on sets of vertices. We prove that for any ϵ > 0, and any set of vertices A that has conductance at most φ, for at least half of the starting vertices in A our algorithm will output (with constant probability) a set of conductance O (√φ /ϵ). We prove that for a given run of the algorithm, the expected ratio between its computational complexity and the volume of the set that it outputs is vol( A ) ϵ φ -1/2 polylog( n ), where vol( A ) = Σ v ∈ A d ( v ) is the volume of the set A . This gives an algorithm with the same guarantee (up to a constant factor) as the Cheeger's inequality that runs in time slightly superlinear in the size of the output. This is the first sublinear (in the size of the input) time algorithm with almost the same guarantee as the Cheeger's inequality. In comparison, the best previous local partitioning algorithm, by Andersen et al. [2006], has a worse approximation guarantee of O (√φ log n ) and a larger ratio of φ -1 polylog( n ) between the complexity and output volume. As a by-product of our results, we prove a bicriteria approximation algorithm for the expansion profile of any graph. For 0 < k ≤ vol( V )/2, let φ( k ) : min S : vol( S ) ≤ k φ( S ). There is a polynomial time algorithm that, for any k , ϵ > 0, finds a set S of volume vol( S ) ≤ O ( k 1 + ϵ ) and expansion φ( S )≤ O (√φ ( k )/ϵ). As a new technical tool, we show that for any set S of vertices of a graph, a lazy t -step random walk started from a randomly chosen vertex of S will remain entirely inside S with probability at least (1 - φ( S )/2) t . This itself provides a new lower bound to the uniform mixing time of any finite state reversible Markov chain.

Frustration index and Cheeger inequalities for discrete and continuous magnetic Laplacians

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.

Cheeger inequalities for unbounded graph Laplacians

We use the concept of intrinsic metrics to give a new definition for an isoperimetric constant of a graph. We use this novel isoperimetric constant to prove a Cheeger-type estimate for the bottom of the spectrum which is nontrivial even if the vertex degrees are unbounded.

Higher dimensional discrete Cheeger inequalities

For graphs there exists a strong connection between spectral and combinatorial expansion properties. This is expressed, e.g., by the discrete Cheeger inequality, the lower bound of which states that $\lambda(G) \leq h(G)$ , where $\lambda(G)$ is the second smallest eigenvalue of the Laplacian of a graph $G$ and $h(G)$ is the Cheeger constant measuring the edge expansion of $G$ . We are interested in generalizations of expansion properties to finite simplicial complexes of higher dimension (or uniform hypergraphs ). Whereas higher dimensional Laplacians were introduced already in 1945 by Eckmann , the generalization of edge expansion to simplicial complexes is not straightforward. Recently, a topologically motivated notion analogous to edge expansion that is based on $\mathbb{Z}_2$ - cohomology was introduced by Gromov and independently by Linial, Meshulam and Wallach . It is known that for this generalization there is no direct higher dimensional analogue of the lower bound of the Cheeger inequality. A different, combinatorially motivated generalization of the Cheeger constant, denoted by $h(X)$ , was studied by Parzanchevski , Rosenthal and Tessler . They showed that indeed $\lambda(X) \leq h(X)$ , where $\lambda(X)$ is the smallest non-trivial eigenvalue of the ( $(k-1)$ -dimensional upper) Laplacian , for the case of $k$ -dimensional simplicial complexes $X$ with complete $(k-1)$ -skeleton. Whether this inequality also holds for $k$ -dimensional complexes with non-com \-plete $(k-1)$ -skeleton has been an open question. We give two proofs of the inequality for arbitrary complexes. The proofs differ strongly in the methods and structures employed, and each allows for a different kind of additional strengthening of the original result.

Multi-way dual Cheeger constants and spectral bounds of graphs

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.

On hyperboundedness and spectrum of Markov operators

Consider an ergodic Markov operator $$M$$ reversible with respect to a probability measure $$\mu $$ on a general measurable space. It is shown that if $$M$$ is bounded from $$\mathbb {L}^2(\mu )$$ to $$\mathbb {L}^p(\mu )$$ , where $$p>2$$ , then it admits a spectral gap. This result answers positively a conjecture raised by Høegh-Krohn and Simon (J. Funct. Anal. 9:121–80, 1972) in the more restricted 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 et al. (Proceedings of the 2012 ACM Symposium on Theory of Computing, 1131–1140, ACM, New York, 2012). 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 and the exponential behaviors of the small eigenvalues of Witten Laplacians at small temperature are recovered.

A Cheeger-type inequality on simplicial complexes

In this paper, we consider a variation on Cheeger numbers related to the coboundary expanders recently defined by Dotterrer and Kahle. A Cheeger-type inequality is proved, which is similar to a result on graphs due to Fan Chung. This inequality is then used to study the relationship between coboundary expanders on simplicial complexes and their corresponding eigenvalues, complementing and extending results found by Gundert and Wagner. In particular, we find these coboundary expanders do not satisfy natural Buser or Cheeger inequalities.

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

Let L be a reversible Markovian generator on a finite set V. Relations between the spectral decomposition of L and subpartitions of the state space V into a given number of components which are optimal with respect to min–max or max–min Dirichlet connectivity criteria are investigated. Links are made with higher-order Cheeger inequalities and with a generic characterization of subpartitions given by the nodal domains of an eigenfunction. These considerations are applied to generators whose positive rates are supported by the edges of a discrete cycle ZN, to obtain a full description of their spectra and of the shapes of their eigenfunctions, as well as an interpretation of the spectrum through a double-covering construction. Also, we prove that for these generators, higher Cheeger inequalities hold, with a universal constant factor 48.

Functional Inequalities for Heavy Tailed Distributions and Application to Isoperimetry

This paper is devoted to the study of probability measures with heavy tails. Using the Lyapunov function approach we prove that such measures satisfy different kind of functional inequalities such as weak Poincaré and weak Cheeger, weighted Poincaré and weighted Cheeger inequalities and their dual forms. Proofs are short and we cover very large situations. For product measures on $\mathbb{R}^n$ we obtain the optimal dimension dependence using the mass transportation method. Then we derive (optimal) isoperimetric inequalities. Finally we deal with spherically symmetric measures. We recover and improve many previous result