Paper Info
Title
A robust khintchine inequality, and algorithms for computing optimal constants in fourier analysis and high-dimensional geometry
Abstract
This paper makes two contributions towards determining some well-studied optimal constants in Fourier analysis of Boolean functions and high-dimensional geometry. 1 It has been known since 1994 [GL94] that every linear threshold function has squared Fourier mass at least 1/2 on its degree-0 and degree-1 coefficients. Denote the minimum such Fourier mass by W≤1[LTF], where the minimum is taken over all n-variable linear threshold functions and all n≥0. Benjamini, Kalai and Schramm [BKS99] have conjectured that the true value of W≤1[LTF] is 2/π. We make progress on this conjecture by proving that W≤1[LTF]≥1/2+c for some absolute constant c0. The key ingredient in our proof is a "robust" version of the well-known Khintchine inequality in functional analysis, which we believe may be of independent interest. 2 We give an algorithm with the following property: given any η0, the algorithm runs in time 2poly(1/η) and determines the value of W≤1[LTF] up to an additive error of ±η. We give a similar 2poly(1/η)-time algorithm to determine Tomaszewski's constant to within an additive error of ±η; this is the minimum (over all origin-centered hyperplanes H) fraction of points in {−1,1}n that lie within Euclidean distance 1 of H. Tomaszewski's constant is conjectured to be 1/2; lower bounds on it have been given by Holzman and Kleitman [HK92] and independently by Ben-Tal, Nemirovski and Roos [BTNR02]. Our algorithms combine tools from anti-concentration of sums of independent random variables, Fourier analysis, and Hermite analysis of linear threshold functions.
Year
DOI
Venue
2013
10.1137/130919143
SIAM J. Discrete Math.
Keywords
Field
DocType
absolute constant c0,linear threshold function,fourier analysis,robust khintchine inequality,hermite analysis,time algorithm,n-variable linear threshold function,additive error,fourier mass,high-dimensional geometry,well-studied optimal constant,functional analysis
Boolean function,Square (algebra),Fourier analysis,Khintchine inequality,Hermite polynomials,Fourier transform,Hyperplane,Geometry,Discrete mathematics,Combinatorics,Algorithm,Physical constant,Mathematics
Conference
Volume
Issue
ISSN
30
2
0895-4801
Citations 
PageRank 
References 
2
0.37
14
Authors
3
Name
Order
Citations
PageRank
Anindya De123924.77
Ilias Diakonikolas277664.21
Rocco A. Servedio31656133.28