Boolean constant degree functions on the slice are juntas

Filmus, Yuval; Ihringer, Ferdinand

doi:10.1016/j.disc.2019.111614

Cited by 13 publications

(35 citation statements)

References 10 publications

(12 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Finally, with a little more representation theory effort, we are able to derive from Theorem 1.1 a Nisan-Szegedy Theorem for the multislice (cf. [48]), which is (roughly) an = 0 version of the Friedgut Junta Theorem; this generalizes previous work on the Hamming slice [25]. It says that if A ⊆ U κ is of "degree k" -meaning that its indicator function can be written as a linear combination of k-junta functions -then A must be an exp(O(k))-junta itself.…”

Section: Applicationssupporting

confidence: 62%

Log-Sobolev inequality for the multislice, with applications

Filmus

O’Donnell

2022

Electron. J. Probab.

Self Cite

View full text Add to dashboard Cite

Let κ ∈ N + satisfy κ1 + • • • + κ = n, and let Uκ denote the multislice of all strings u ∈ [ ] n having exactly κi coordinates equal to i, for all i ∈ [ ]. Consider the Markov chain on Uκ where a step is a random transposition of two coordinates of u. We show that the log-Sobolev constant κ for the chain satisfieswhich is sharp up to constants whenever is constant. From this, we derive some consequences for small-set expansion and isoperimetry in the multislice, including a KKL Theorem, a Kruskal-Katona Theorem for the multislice, a Friedgut Junta Theorem, and a Nisan-Szegedy Theorem.

show abstract

Section: Applicationssupporting

confidence: 62%

Log-Sobolev inequality for the multislice, with applications

Filmus

O’Donnell

2022

Electron. J. Probab.

Self Cite

View full text Add to dashboard Cite

show abstract

“…Classical results in analysis of Boolean functions on the hypercube show that Ω(2 d ) ≤ γ d (H(n, 2)) ≤ d2 d−1 . In ongoing work [28], we have shown that for k, n − k ≥ exp(d) it holds that γ d (J(n, k)) = γ d (H(n, 2)), and we suspect that a similar result holds for the multislice. More generally, we conjecture that in the domains considered in the paper, Boolean degree d functions can be formed by combining a bounded number of Boolean degree 1 functions.…”

Section: Future Workmentioning

confidence: 83%

Boolean degree 1 functions on some classical association schemes

Filmus

Ihringer

2019

Journal of Combinatorial Theory, Series A

Self Cite

View full text Add to dashboard Cite

We investigate Boolean degree 1 functions for several classical association schemes, including Johnson graphs, Grassmann graphs, graphs from polar spaces, and bilinear forms graphs, as well as some other domains such as multislices (Young subgroups of the symmetric group). In some settings, Boolean degree 1 functions are also known as completely regular strength 0 codes of covering radius 1, Cameron-Liebler line classes, and tight sets.We classify all Boolean degree 1 functions on the multislice. On the Grassmann scheme Jq(n, k) we show that all Boolean degree 1 functions are trivial for n ≥ 5, k, n − k ≥ 2 and q ∈ {2, 3, 4, 5}, and that, for general q, the problem can be reduced to classifying all Boolean degree 1 functions on Jq(n, 2). We also consider polar spaces and the bilinear forms graphs, giving evidence that all Boolean degree 1 functions are trivial for appropriate choices of the parameters.1 (also strength 0 designs) [44]. Motivated by problems on permutation groups and finite geometry, Boolean degree 1 functions are also known as tight sets [2,12] and Cameron-Liebler line classes [17]. The history of Cameron-Liebler line classes is particularly complicated, as the problem was introduced by Cameron and Liebler [7], the term coined by Penttila [52,53], and the algebraic point of view as Boolean degree 1 functions only emerged later; see [59], in particular §3.3.1, for a discussion of this.Due to this variation of terminology, classification results of Boolean degree 1 functions on J(n, k) were obtained repeatedly in the literature (with small variations due to different definitions) at least three times, in [49] for completely regular strength 0 codes of covering radius 1, in [25] for Boolean degree 1 functions, and in [11] for Cameron-Liebler line classes:Theorem 1.2 (Folklore). Suppose that k, n − k ≥ 2. Every Boolean degree 1 function on J(n, k) is either constant or depends on a single coordinate.Depending on the definition used, this is either easy to observe or requires a more elaborate proof. For the hypercube H(n, 2), which is a product domain, and the Johnson graph J(n, k) classifying Boolean degree 1 functions is trivial, but there are various other classical association schemes for which classification is more difficult. The Grassmann scheme J q (n, k) consists of all k-spaces of F n q as vertices, two vertices being adjacent if their meet is a subspace of dimension k − 1. Boolean degree 1 functions on J q (4, 2) were intensively investigated, and many non-trivial examples [6,8,9,24,33,37] and existence conditions [34,47] are known.We call 1-dimensional subspaces of F n q points, 2-dimensional subspaces of F n q lines, and (n − 1)dimensional subspaces of F n q hyperplanes. For a point p we define p + (S) = 1 p∈S and p − (S) = 1 p / ∈S , and for a hyperplane π we define π + (S) = 1 S⊆π and π − (S) = 1 S π . The following was shown by Drudge for q = 3 [17, Theorem 6.4]; by Gavrilyuk and Mogilnykh for q = 4 [35, Theorem 3]; and by Gavrilyuk and Matkin [32,45] for q = 5; the result for q = 2 fol...

show abstract

“…[FI18] recently considered an analog of the parameter R(f ) for the family of level k slice functions which are Boolean functions whose domain is restricted to the set of inputs of Hamming weight exactly k. They showed that, provided that min(k, n−k) is sufficiently large (at least B d for some fixed constant B) then every level k slice function on n-variables of degree at most d depends on at most R d variables. (See [FI18] for precise definitions and details.) Thus our improved upper bound on R d applies also to the number of relevant variables of slice functions.…”

Section: Introductionmentioning

confidence: 99%

An Asymptotically Tight Bound on the Number of Relevant Variables in a Bounded Degree Boolean function

2020

View full text Add to dashboard Cite

We prove that there is a constant C ≤ 6.614 such that every Boolean function of degree at most d (as a polynomial over R) is a C ·2 d -junta, i.e. it depends on at most C ·2 d variables. This improves the d · 2 d−1 upper bound of Nisan and Szegedy [Computational Complexity 4 (1994)].The bound of C · 2 d is tight up to the constant C as a lower bound of 2 d − 1 is achieved by a read-once decision tree of depth d. We slightly improve the lower bound by constructing, for each positive integer d, a function of degree d with 3 · 2 d−1 − 2 relevant variables. A similar construction was independently observed by Shinkar and Tal.

show abstract

Boolean constant degree functions on the slice are juntas

Cited by 13 publications

References 10 publications

Log-Sobolev inequality for the multislice, with applications

Log-Sobolev inequality for the multislice, with applications

Boolean degree 1 functions on some classical association schemes

An Asymptotically Tight Bound on the Number of Relevant Variables in a Bounded Degree Boolean function

Contact Info

Product

Resources

About