Local Spectral Expansion Approach to High Dimensional Expanders Part I: Descent of Spectral Gaps

Oppenheim, Izhar

doi:10.1007/s00454-017-9948-x

Cited by 69 publications

(78 citation statements)

References 12 publications

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“…These matrices generalize the classical graph Laplacian and there has been extensive research to study their eigenvalues [see Lub17, and the references therein]. A method known as Garland's method [Gar73] relates the eigenvalues of graph Laplacians of 1-skeletons of links of X to eigenvalues of high dimensional Laplacians of X [see BŚ97;Opp18].…”

Section: Techniquesmentioning

confidence: 99%

Log-concave polynomials II: high-dimensional walks and an FPRAS for counting bases of a matroid

Anari

Liu

Gharan

et al. 2019

Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing

105

136

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We design an FPRAS to count the number of bases of any matroid given by an independent set oracle, and to estimate the partition function of the random cluster model of any matroid in the regime where 0 < q < 1. Consequently, we can sample random spanning forests in a graph and (approximately) compute the reliability polynomial of any matroid. We also prove the thirty year old conjecture of Mihail and Vazirani that the bases exchange graph of any matroid has expansion at least 1.Our algorithm and the proof build on the recent results of Dinur, Kaufman, Mass and Oppenheim [KM17; DK17; KO18] who show that high dimensional walks on simplicial complexes mix rapidly if the corresponding localized random walks on 1-skeleton of links of all complexes are strong spectral expanders. One of our key observations is a close connection between pure simplicial complexes and multiaffine homogeneous polynomials. Specifically, if X is a pure simplicial complex with positive weights on its maximal faces, we can associate with X a multiaffine homogeneous polynomial p X such that the eigenvalues of the localized random walks on X correspond to the eigenvalues of the Hessian of derivatives of p X .

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Section: Techniquesmentioning

confidence: 99%

Log-concave polynomials II: high-dimensional walks and an FPRAS for counting bases of a matroid

Anari

Liu

Gharan

et al. 2019

Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing

105

136

View full text Add to dashboard Cite

show abstract

“…This method, proves global properties of the simplicial complexes by properties on the links. This method, originally developed by Garland in [Gar73], is used in many works such as [EK16,DK17,Opp18a].…”

Section: The Complement Random Walk In High Dimensional Expandersmentioning

confidence: 99%

“…We now go towards proving Lemma 7.5, since its corollary, Corollary 7.6 is the base case for proving Theorem 7.1, item 2. This lemma is an adaptation of the theorem in [Opp18a], where the author proved the following:…”

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confidence: 94%

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Agreement Testing Theorems on Layered Set Systems

Dikstein

Dinur

2019

2019 IEEE 60th Annual Symposium on Foundations of Computer Science (FOCS)

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We introduce a framework of layered subsets, and give a sufficient condition for when a set system supports an agreement test. Agreement testing is a certain type of property testing that generalizes PCP tests such as the plane vs. plane test. Previous work has shown that high dimensional expansion is useful for agreement tests. We extend these results to more general families of subsets, beyond simplicial complexes. These include -Agreement tests for set systems whose sets are faces of high dimensional expanders. Our new tests apply to all dimensions of complexes both in case of two-sided expansion and in the case of one-sided partite expansion. This improves and extends an earlier work of Dinur and Kaufman (FOCS 2017) and applies to matroids, and potentially many additional complexes.-Agreement tests for set systems whose sets are neighborhoods of vertices in a high dimensional expander. This family resembles the expander neighborhood family used in the gap-amplification proof of the PCP theorem. This set system is quite natural yet does not sit in a simplicial complex, and demonstrates some versatility in our proof technique.-Agreement tests on families of subspaces (also known as the Grassmann poset). This extends the classical low degree agreement tests beyond the setting of low degree polynomials.Our analysis relies on a new random walk on simplicial complexes which we call the "complement random walk" and which may be of independent interest. This random walk generalizes the non-lazy random walk on a graph to higher dimensions, and has significantly better expansion than previously-studied random walks on simplicial complexes.

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“…Unlike the graph case d = 1, where various different notions of expansion are closely related, in the high-dimensional case d > 1 the several notions of expansion that have been proposed in the literature are in general far from being equivalent, and the relationship between them is still poorly understood. Notions of expansion for d > 1 that have been proposed include the aforementioned spectral expansion [25,27,30,52], combinatorial expansion [8,26,29,53,55], geometric and topological expansion [13,20,21,28,50], F 2 -coboundary expansion [12,28,30,42,43,49,57] and Ramanujan complexes [11,15,27,37,41,46]. In addition, the adjacency matrix can be interpreted as a generator of a stochastic process which is a high-dimensional analog of simple random walks, see [48,54,56] for more details.…”

Section: Introductionmentioning

confidence: 99%

Eigenvalue confinement and spectral gap for random simplicial complexes

Knowles

Rosenthal

2017

Random Struct Algorithms

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We consider the adjacency operator of the Linial-Meshulam model for random simplicial complexes on n vertices, where each d-cell is added independently with probability p to the complete (d − 1)-skeleton. Under the assumption np(1 − p) log 4 n, we prove that the spectral gap between the n−1 d smallest eigenvalues and the remaining (1)) with high probability. This estimate follows from a more general result on eigenvalue confinement. In addition, we prove that the global distribution of the eigenvalues is asymptotically given by the semicircle law. The main ingredient of the proof is a Füredi-Komlós-type argument for random simplicial complexes, which may be regarded as sparse random matrix models with dependent entries.

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Local Spectral Expansion Approach to High Dimensional Expanders Part I: Descent of Spectral Gaps

Cited by 69 publications

References 12 publications

Log-concave polynomials II: high-dimensional walks and an FPRAS for counting bases of a matroid

Log-concave polynomials II: high-dimensional walks and an FPRAS for counting bases of a matroid

Agreement Testing Theorems on Layered Set Systems

Eigenvalue confinement and spectral gap for random simplicial complexes

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