High Dimensional Expanders: Eigenstripping, Pseudorandomness, and Unique Games

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

doi:10.48550/arxiv.2011.04658

Cited by 1 publication

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“…Fourier Analysis on HDX: High dimensional expanders (HDX) are a class of robustly connected complexes that have seen an incredible amount of development and application throughout theoretical computer science in the past few years, most famously in coding theory [DEL + 21, EKZ20, JST21, KO21, KT21b, DDHRZ20, JQST20, DHK + 19] and approximate sampling [ALOV19, AL20, ALO20, CLV20, CLV21, CGŠV21, FGYZ21, JPV21, Liu21, BCC + 21], but also in agreement testing [DK17, DD19, KM20], CSPapproximation [AJT19,BHKL20], and (implicitly) hardness of approximation [KMMS18,KMS18]. In this work, we study a central notion of high dimensional expansion called two-sided local-spectral expansion, originally developed by Dinur and Kaufman [DK17] to build sparse agreement testers.…”

Section: Contributionsmentioning

confidence: 99%

“…2 Traditionally proved via hypercontractivity, a variant of this result on the Grassmannian recently led to the resolution of the 2-2 Games Conjecture [KMS18]. On the other hand, Bafna, Hopkins, Kaufman, and Lovett [BHKL20] showed that second moment methods cannot recover such a result. While they are able to recover some sort of characterization with these techniques, it necessarily decays as the dimension grows to infinity, becoming trivial in the regime useful for hardness of approximation-if we want to do better, it appears we need a theory of hypercontractivity.…”

Section: Contributionsmentioning

confidence: 99%

“…4 Combined with BHKL's recent spectral analysis of higher order random walks (which, for the moment, we'll think of as analogs of the noisy hypercube graphs or Hamming scheme), this leads to the resolution of a number of open questions in boolean function analysis. To start, we provide a tight characterization of (edge) expansion on higher order random walks, which, unlike previous methods [BHKL20], does not decay with dimension. This matches the version of the result on the Grassmannian which led to the resolution of the 2-2 Games Conjecture [KMS18], and opens yet another avenue towards the use of HDX in hardness of approximation.…”

Section: Contributionsmentioning

confidence: 99%

“…A similar decomposition was also proposed around the same time by Kaufman and Oppenheim [KO20], though their work was more focused on understanding the spectral structure of higher order random walks than on developing a theory of Fourier analysis. In the years since, the HD-Level-Set Decomposition has seen some further development [AJT19, KS20, BHKL20], and the nascent theory has helped build efficient approximation algorithms for certain k-CSPs [AJT19] and unique games [BHKL20], but the restriction to second moment methods seems to have limited its use otherwise. Towards breaking this same barrier, Gur, Lifshitz, and Liu [GLL21] have also (independently) developed a similar theory of hypercontractivity on local-spectral expanders.…”

Section: Related Workmentioning

confidence: 99%

“…A basic inductive argument then implies that D k i−1 U k i g ↑i ≤ 2 O(i) k 2 γ f , which completes the proof. For a more formal induction following exactly the same strategy, see [DDFH18,BHKL20].…”

Section: Bottom-up Vs Hd-level-setmentioning

confidence: 99%

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Hypercontractivity on High Dimensional Expanders: a Local-to-Global Approach for Higher Moments

Bafna¹,

Hopkins²,

Kaufman³

et al. 2021

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Hypercontractivity is one of the most powerful tools in Boolean function analysis. Originally studied over the discrete hypercube, recent years have seen increasing interest in extensions to settings like the p-biased cube, slice, or Grassmannian, where variants of hypercontractivity have found a number of breakthrough applications including the resolution of Khot's 2-2 Games Conjecture (Khot, Minzer, Safra FOCS 2018). In this work, we develop a new theory of hypercontractivity on high dimensional expanders (HDX), an important class of expanding complexes that has recently seen similarly impressive applications in both coding theory and approximate sampling. Our results lead to a new understanding of the structure of Boolean functions on HDX, including a tight analog of the KKL Theorem and a new characterization of non-expanding sets.Unlike previous settings satisfying hypercontractivity, HDX can be asymmetric, sparse, and very far from products, which makes the application of traditional proof techniques challenging. We handle these barriers with the introduction of two new tools of independent interest: a new explicit combinatorial Fourier basis for HDX that behaves well under restriction, and a new local-to-global method for analyzing higher moments. Interestingly, unlike analogous second moment methods that apply equally across all types of expanding complexes, our tools rely inherently on simplicial structure. This suggests a new distinction among high dimensional expanders based upon their behavior beyond the second moment.

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Section: Contributionsmentioning

confidence: 99%

Section: Contributionsmentioning

confidence: 99%

Section: Contributionsmentioning

confidence: 99%

Section: Related Workmentioning

confidence: 99%

Section: Bottom-up Vs Hd-level-setmentioning

confidence: 99%

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