High Dimensional Expanders: Eigenstripping, Pseudorandomness, and Unique Games

Bafna, Mitali; Hopkins, Max; Kaufman, Tali; Lovett, Shachar

doi:10.1137/1.9781611977073.47

Cited by 18 publications

(41 citation statements)

References 37 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…This generalizes and tightens recent work on expanding hypergraphs of BHKL [15,Theorem 2.2] (which itself extended a number earlier works on the topic [2,3,14]).…”

Section: Introductionsupporting

confidence: 82%

“…As such, our main focus in this work lies in characterizing the spectral and combinatorial behavior of random walks on eposets beyond the second eigenvalue. Strengthening DDFH and recent work of Bafna, Hopkins, Kaufman, and Lovett (BHKL) [15], we prove that at a coarse level (walks on) eposets indeed exhibit the same spectral and combinatorial characteristics as expanding hypergraphs (e.g. spectral stripping, expansion of pseudorandom sets).…”

Section: Introductionsupporting

confidence: 52%

“…Random walks on high dimensional expanders (HDX) have been the object of intense study in theoretical computer science in recent years. Starting with their original formulation by Kaufman and Mass [1], a series of works on the spectral structure of these walks [2][3][4] led to significant breakthroughs in approximate sampling [5,4,[6][7][8][9][10][11][12][13], CSP-approximation [14,15], error-correcting codes [16,17], agreement testing [18][19][20], and more. Most of these works focus on the structure of expansion in hypergraphs (typically called simplicial complexes in the HDX literature).…”

Section: Introductionmentioning

confidence: 99%

“…In slightly more detail, we start by showing that all eposets exhibit a behaviour called "eigenstripping" [2,3,15]: the spectrum of any associated random walk concentrates around a few unique approximate eigenvalues.…”

Section: Introductionmentioning

confidence: 99%

“…for hardness of approximation [25] or fast algorithms [15]), and while it is possible to recover strong decay on weaker posets by increasing the length of the walk [15], this is usually untenable in application due to the additional degrees of freedom it affords. 3 The spectral structure of walks on eposets is closely related to their edge-expansion, an important combinatorial property that has recently played a crucial role both in algorithms for [33,15] and hardness of unique games [25]. The key insight in both cases lay in understanding the structure of non-expanding sets.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Eigenstripping, Spectral Decay, and Edge-Expansion on Posets

Gaitonde¹,

Hopkins²,

Kaufman³

et al. 2022

Preprint

Self Cite

View full text Add to dashboard Cite

Fast mixing of random walks on hypergraphs (simplicial complexes) has led to myriad breakthroughs throughout theoretical computer science in the last five years. On the other hand, many important applications (e.g. to locally testable codes, 2-2 games) rely on a more general class of underlying structures called posets, and crucially take advantage of non-simplicial structure. These works make it clear that the global expansion properties of posets depend strongly on their underlying architecture (e.g. simplicial, cubical, linear algebraic), but the overall phenomenon remains poorly understood. In this work, we quantify the advantage of different poset architectures in both a spectral and combinatorial sense, highlighting how regularity controls the spectral decay and edge-expansion of corresponding random walks.We show that the spectra of walks on expanding posets (Dikstein, Dinur, Filmus, Harsha APPROX-RANDOM 2018) concentrate in strips around a small number of approximate eigenvalues controlled by the regularity of the underlying poset. This gives a simple condition to identify poset architectures (e.g. the Grassmann) that exhibit strong (even exponential) decay of eigenvalues, versus architectures like hypergraphs whose eigenvalues decay linearly-a crucial distinction in applications to hardness of approximation and agreement testing such as the recent proof of the 2-2 Games Conjecture (Khot, Minzer, Safra FOCS 2018). We show these results lead to a tight characterization of edge-expansion on expanding posets in the ℓ2-regime (generalizing recent work of Bafna, Hopkins, Kaufman, and Lovett (SODA 2022)), and pay special attention to the case of the Grassmann where we show our results are tight for a natural set of sparsifications of the Grassmann graphs. We note for clarity that our results do not recover the characterization of expansion used in the proof of the 2-2 Games Conjecture which relies on ℓ8 rather than ℓ2-structure.

show abstract

“…This generalizes and tightens recent work on expanding hypergraphs of BHKL [15,Theorem 2.2] (which itself extended a number earlier works on the topic [2,3,14]).…”

Section: Introductionsupporting

confidence: 82%

Section: Introductionsupporting

confidence: 52%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Eigenstripping, Spectral Decay, and Edge-Expansion on Posets

Gaitonde¹,

Hopkins²,

Kaufman³

et al. 2022

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

Near coverings and cosystolic expansion

Dinur

Meshulam

2022

Arch. Math.

View full text Add to dashboard Cite

We introduce and study swap cosystolic expansion, a new expansion property of simplicial complexes. We prove lower bounds for swap coboundary expansion of spherical buildings and use them to lower bound swap cosystolic expansion of the LSV Ramanujan complexes. Our motivation is the recent work (in a companion paper) showing that swap cosystolic expansion implies agreement theorems. Together the two works show that these complexes support agreement tests in the low acceptance regime.Swap cosystolic expansion is defined by considering, for a given complex X, its faces complex F r X, whose vertices are r-faces of X and where two vertices are connected if their disjoint union is also a face in X. The faces complex F r X is a derandomizetion of the product of X with itself r times. The graph underlying F r X is the swap walk of X, known to have excellent spectral expansion. The swap cosystolic expansion of X is defined to be the cosystolic expansion of F r X.Our main result is a exp(−O( √ r)) lower bound on the swap coboundary expansion of the spherical building and the swap cosystolic expansion of the LSV complexes. For more general coboundary expanders we show a weaker lower bound of exp(−O(r)).

show abstract

Hypercontractivity for global functions and sharp thresholds

Keevash¹,

Lifshitz²,

Long³

et al. 2023

J. Amer. Math. Soc.

View full text Add to dashboard Cite

The classical hypercontractive inequality for the noise operator on the discrete cube plays a crucial role in many of the fundamental results in the Analysis of Boolean functions, such as the Kahn-Kalai-Linial theorem, Friedgut’s junta theorem and the invariance principle of Mossel, O’Donnell and Oleszkiewicz. In these results the cube is equipped with the uniform ( 1 / 2 1/2 -biased) measure, but it is desirable, particularly for applications to the theory of sharp thresholds, to also obtain such results for general p p -biased measures. However, simple examples show that when p p is small there is no hypercontractive inequality that is strong enough for such applications. In this paper, we establish an effective hypercontractivity inequality for general p p that applies to ‘global functions’, i.e. functions that are not significantly affected by a restriction of a small set of coordinates. This class of functions appears naturally, e.g. in Bourgain’s sharp threshold theorem, which states that such functions exhibit a sharp threshold. We demonstrate the power of our tool by strengthening Bourgain’s theorem, making progress on two conjectures of Kahn and Kalai (both these conjectures were open when we arXived this paper in 2019; one of them was solved in 2022; the other is still open), and proving a p p -biased analogue of the seminal invariance principle of Mossel, O’Donnell, and Oleszkiewicz. In this 2023 version of our paper we will also survey many further applications of our results that have been obtained by various authors since we arXived the first version in 2019.

show abstract

High Dimensional Expanders: Eigenstripping, Pseudorandomness, and Unique Games

Cited by 18 publications

References 37 publications

Eigenstripping, Spectral Decay, and Edge-Expansion on Posets

Eigenstripping, Spectral Decay, and Edge-Expansion on Posets

Near coverings and cosystolic expansion

Hypercontractivity for global functions and sharp thresholds

Contact Info

Product

Resources

About