Log-Sobolev inequality for the multislice, with applications

Filmus, Yuval; O’Donnell, Ryan; Wu, Xinyu

doi:10.1214/22-ejp749

Cited by 11 publications

(12 citation statements)

References 49 publications

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“…This mapping also clearly preserves norms, so f =d 2 = f =d 2 . Our claim will now easily follow from the results of [24]. To be more precise, defining the influence of g : S n → R on a transposition π i,j ∈ [n] as I i,j [g] = Eπ (g(π i,j π) − g(π)) 2 , we see that…”

Section: The Total Noisy Influence Is Constantmentioning

confidence: 86%

An Invariance Principle for the Multi-slice, with Applications

Braverman¹,

Khot²,

Lifshitz³

et al. 2021

Preprint

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Given an alphabet size m ∈ N thought of as a constant, and k = (k 1 , . . . , k m ) whose entries sum of up n, the k-multi-slice is the set of vectors x ∈ [m] n in which each symbol i ∈ [m] appears precisely k i times. We show an invariance principle for low-degree functions over the multi-slice, to functions over the product space ([m] n , µ n ) in which µ(i) = k i /n. This answers a question raised by [22].As applications of the invariance principle, we show:1. An analogue of the "dictatorship test implies computational hardness" paradigm for problems with perfect completeness, for a certain class of dictatorship tests. Our computational hardness is proved assuming a recent strengthening of the Unique-Games Conjecture, called the Rich 2-to-1 Games Conjecture.Using this analogue, we show that assuming the Rich 2-to-1 Games Conjecture, (a) there is an r-ary CSP P r for which it is NP-hard to distinguish satisfiable instances of the CSP and instances that are at most 2r+1 2 r + o(1) satisfiable, and (b) hardness of distinguishing 3-colorable graphs, and graphs that do not contain an independent set of size o(1). A reduction of the problem of studying expectations of products of functions on the multi-slice tostudying expectations of products of functions on correlated, product spaces. In particular, we are able to deduce analogues of the Gaussian bounds from [40] for the multi-slice.3. In a companion paper, we show further applications of our invariance principle in extremal combinatorics, and more specifically to proving removal lemmas of a wide family of hypergraphs H called ζ-forests, which is a natural extension of the well-studied case of matchings.

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Section: The Total Noisy Influence Is Constantmentioning

confidence: 86%

An Invariance Principle for the Multi-slice, with Applications

Braverman¹,

Khot²,

Lifshitz³

et al. 2021

Preprint

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show abstract

“…Recently, Theorem 1.4 has been extended to several other structures, for instance the multislice [25] by Filmus, O'Donnell and Wu, and to the perfect matching scheme by Dafni, Filmus, Lifshitz, Lindzey, and Vinayls [12].…”

Section: Low Degree Boolean Functionsmentioning

confidence: 99%

Degree 2 Boolean Functions on Grassmann Graphs

Beule¹,

D'haeseleer²,

Ihringer³

et al. 2022

Preprint

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We investigate Boolean degree d functions on the Grassmann Graph of k-spaces in F n q . Our focus are degree 2 functions for (n, k) = (6, 3) and (n, k) = (8, 4).We also discuss connections to the analysis of Boolean functions, regular sets/equitable bipartitions/perfect 2-colorings in graphs, q-analogs of designs, and permutation groups. In particular, this represents a natural generalization of Cameron-Liebler line classes.

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“…where µ(S) is the measure of S. We upper-bound this quantity using spectral considerations. We will only need our hypercontractive inequality and basic knowledge of the eigenvalues of T, which can be found, for example, in [10,Corollary 21]. This is the content of the firs three items in the lemma below (we also prove a fourth item, which will be useful for us later on).…”

Section: Isoperimetric Inequalities In the Transpositions Cayley Graphmentioning

confidence: 99%

Hypercontractivity on the symmetric group

Filmus,

Kindler,

Lifshitz

et al. 2020

Preprint

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The hypercontractive inequality is a fundamental result in analysis, with many applications throughout discrete mathematics, theoretical computer science, combinatorics and more. So far, variants of this inequality have been proved mainly for product spaces, which raises the question of whether analogous results hold over non-product domains.We consider the symmetric group, S n , one of the most basic non-product domains, and establish hypercontractive inequalities on it. Our inequalities are most effective for the class of global functions on S n , which are functions whose 2-norm remains small when restricting O(1) coordinates of the input, and assert that low-degree, global functions have small q-norms, for q > 2.As applications, we show:1. An analog of the level-d inequality on the hypercube, asserting that the mass of a global function on low-degrees is very small. We also show how to use this inequality to bound the size of global, product-free sets in the alternating group A n .2. Isoperimetric inequalities on the transposition Cayley graph of S n for global functions, that are analogous to the KKL theorem and to the small-set expansion property in the Boolean hypercube.3. Hypercontractive inequalities on the multi-slice, and stability versions of the Kruskal-Katona Theorem in some regimes of parameters.

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Log-Sobolev inequality for the multislice, with applications

Cited by 11 publications

References 49 publications

An Invariance Principle for the Multi-slice, with Applications

An Invariance Principle for the Multi-slice, with Applications

Degree 2 Boolean Functions on Grassmann Graphs

Hypercontractivity on the symmetric group

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