Inverse Expander Mixing for Hypergraphs

Cohen, Emma; Mubayi, Dhruv; Ralli, Peter; Tetali, Prasad

doi:10.37236/5283

Cited by 10 publications

(10 citation statements)

References 21 publications

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“…shows that pseudo-randomness and two-sided spectral concentration are (almost) equivalent. For complexes with a complete skeleton, a converse to the mixing lemma from [37] was recently established in [8]. Do these converse theorems admit a generalization to the general case?…”

Section: Questionsmentioning

confidence: 99%

Mixing in High-Dimensional Expanders

Parzanchevski¹

2017

Combinator. Probab. Comp.

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We prove a generalization of the Expander Mixing Lemma for arbitrary (finite) simplicial complexes. The original lemma states that concentration of the Laplace spectrum of a graph implies combinatorial expansion (which is also referred to as mixing, or quasi-randomness).Recently, an analogue of this Lemma was proved for simplicial complexes of arbitrary dimension, provided that the skeleton of the complex is complete. More precisely, it was shown that a concentrated spectrum of the simplicial Hodge Laplacian implies a similar type of expansion as in graphs. In this paper we remove the assumption of a complete skeleton, showing that concentration of the Laplace spectra in all dimensions implies combinatorial expansion in any complex. As applications we show that spectral concentration implies Gromov's geometric overlap property, and can be used to bound the chromatic number of a complex ( †) Supported by The Fund for Math at the Institute for Advanced Study.

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Section: Questionsmentioning

confidence: 99%

Mixing in High-Dimensional Expanders

Parzanchevski¹

2017

Combinator. Probab. Comp.

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“…Several results on hypergraph expansion have been obtained using the Friedman-Wigderson approach. Hyperedge expansion depending on the spectral norm of the associated tensor was studied in the original paper [18], the relation between the spectral gap and quasirandom properties was discussed in Lenz and Mubayi [22,23], and an inverse expander mixing lemma was obtained in Cohen et al [12]. Very recently, Li and Mohar [24] proved a generalization of the Alon-Boppana bound to (d, k)-regular hypergraphs for their adjacency tensors.…”

Section: Introductionmentioning

confidence: 99%

Spectra of Random Regular Hypergraphs

Dumitriu

Zhu

2021

Electron. J. Combin.

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In this paper, we study the spectra of regular hypergraphs following the definitions from Feng and Li (1996). Our main result is an analog of Alon's conjecture for the spectral gap of the random regular hypergraphs. We then relate the second eigenvalues to both its expansion property and the mixing rate of the non-backtracking random walk on regular hypergraphs. We also prove the spectral gap for the non-backtracking operator of a random regular hypergraph introduced in Angelini et al. (2015). Finally, we obtain the convergence of the empirical spectral distribution (ESD) for random regular hypergraphs in different regimes. Under certain conditions, we can show a local law for the ESD.

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“…There are several hypergraph expander mixing lemmas in the literature based on the spectral norm of tensors [27,43,19]. However, for deterministic tensors, the spectral norm is NP-hard to compute [33], hence those estimates might not be applicable in practice.…”

Section: Sparse Hypergraph Expandersmentioning

confidence: 99%

Sparse random tensors: Concentration, regularization and applications

Zhu

2021

Electron. J. Statist.

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We prove a non-asymptotic concentration inequality for the spectral norm of sparse inhomogeneous random tensors with Bernoulli entries. For an order-k inhomogeneous random tensor T with sparsity pmax ≥) with high probability. The optimality of this bound up to polylog factors is provided by an information theoretic lower bound. By tensor unfolding, we extend the range of sparsity to pmax ≥ c log n n m with 1 ≤ m ≤ k − 1 and obtain concentration inequalities for different sparsity regimes. We also provide a simple way to regularize T such that O( √ n m pmax) concentration still holds down to sparsity pmax ≥ c n m with k/2 ≤ m ≤ k − 1. We present our concentration and regularization results with two applications: (i) a randomized construction of hypergraphs of bounded degrees with good expander mixing properties, (ii) concentration of sparsified tensors under uniform sampling.

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Inverse Expander Mixing for Hypergraphs

Abstract: We formulate and prove inverse mixing lemmas in the settings of simplicial complexes and k-uniform hypergraphs. In the hypergraph setting, we extend results of Bilu and Linial for graphs. In the simplicial complex setting, our results answer a question of Parzanchevski et al.

Cited by 10 publications

References 21 publications

Mixing in High-Dimensional Expanders

Mixing in High-Dimensional Expanders

Spectra of Random Regular Hypergraphs

Sparse random tensors: Concentration, regularization and applications

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