Local Spectral Expansion Approach to High Dimensional Expanders Part II: Mixing and Geometrical Overlapping

Oppenheim, Izhar

doi:10.1007/s00454-019-00117-7

Cited by 4 publications

(5 citation statements)

References 14 publications

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“…These walks generalize the non-lazy adjacency operator in a graph, and the bipartite adjacency operator in a bipartite graph to high dimensions. As a bonus we show an immediate application for these walks: a new high dimensional expander mixing lemma for sets in all dimensions (see Lemma 7.14 and Lemma 7.15), extending the work of [LGE15,Opp18b].…”

Section: The Complement Random Walk In High Dimensional Expandersmentioning

confidence: 71%

“…In an exciting recent work [ALGV18] it was proven that this complex is a 0-one-sided HDX. Oppenheim [Opp18b] proved that if we truncate this complex by keeping only faces of dimensions 0 i d then it becomes a 1/(r − d − 2)-two-sided HDX. We reach the following conclusion r, the collection of independent sets in a matroid whose size is d supports a sound agreement test.…”

Section: Agreement On High Dimensional Expandersmentioning

confidence: 99%

“…There are other suggested expander mixing lemmas for high dimensional expanders. For example, the lemma in [Opp18b] states that on a λ-two-sided high dimensional expander, for A 1 , ..., A m ⊂ X(0) we get that…”

Section: Random Walks With Fixed Union Sizementioning

confidence: 99%

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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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Section: The Complement Random Walk In High Dimensional Expandersmentioning

confidence: 71%

Section: Agreement On High Dimensional Expandersmentioning

confidence: 99%

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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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“…This result has several applications to random topology. Indeed, Garland's method [Gar73] and its refinements (see [ Ż96], [ Ż03], [Opp18], [Opp20]), Żuk's criterion among them, have proven to be very effective tools for extracting global information about a pure k-dimensional simplicial complex using only information found in the k − 2-dimensional links of the complex.…”

Section: Applications To Random Topologymentioning

confidence: 99%

Determinantal Hypertrees

Werf¹

2022

Preprint

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We deduce a structurally inductive description of Kalai's famous tree generalization, the Q-acyclic complex, which extends to an analogous description of the determinantal probability measure associated with Kalai's celebrated enumeration result. Along the way, we generalize the homological notion of a simplicial complex being Q-acyclic to a relative-homological notion of a pair of simplicial complexes being Q-acyclic relative to each other. We also apply these new results to random topology and the spectral analysis of random graphs.

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“…We have avoided this notation to (i) prevent potential confusion with X[j] and (ii) to avoid refering to an n-partite complex as an (n − 1)-dimensional n-partite complex, as was done in e.g. [Opp18,DD19].…”

Section: (Partite) Simplicial Complexesmentioning

confidence: 99%

Improved analysis of higher order random walks and applications

Alev

Lau

2020

Proceedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing

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We study a variant of the down-up (also known as the Glauber dynamics) and up-down walks over an n-partite simplicial complex, which we call expanderized higher order random walks -where the sequence of updated coordinates correspond to the sequence of vertices visited by a random walk over an auxiliary expander graph H. When H is the clique with self loops on [n], this random walk reduces to the usual down-up walk and when H is the directed cycle on [n], this random walk reduces to the well-known systematic scan Glauber dynamics. We show that whenever the usual higher order random walks satisfy a log-Sobolev inequality or a Poincaré inequality, the expanderized walks satisfy the same inequalities with a loss of quality related to the two-sided expansion of the auxillary graph H. Our construction can be thought as a higher order random walk generalization of the derandomized squaring algorithm of Rozenman and Vadhan (RANDOM 2005).We study the mixing times of our expanderized walks in two example cases: We show that when initiated with an expander graph our expanderized random walks have mixing time (i) O(n log n) for sampling a uniformly random list colorings of a graph G of maximum degree ∆ = O(1) where each vertex has at least (11/6 − ε)∆ and at most O(∆) colors, (ii) O h n log n

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Local Spectral Expansion Approach to High Dimensional Expanders Part II: Mixing and Geometrical Overlapping

Cited by 4 publications

References 14 publications

Agreement Testing Theorems on Layered Set Systems

Agreement Testing Theorems on Layered Set Systems

Determinantal Hypertrees

Improved analysis of higher order random walks and applications

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