A Robust Khintchine Inequality, and Algorithms for Computing Optimal Constants in Fourier Analysis and High-Dimensional Geometry

This paper makes two contributions towards determining some well-studied optimal constants in Fourier analysis of Boolean functions and high-dimensional geometry. It has been known since 1994 [C. Gotsman and N. Linial, Combinatorica, 14 (1994), pp. 35--50] that every linear threshold function (LTF) has a squared Fourier mass of at least $1/2$ on its degree-$0$ and degree-$1$ coefficients. Let the minimum such Fourier mass be ${\bf W}^{\leq 1}[{\bf LTF}]$, where the minimum is taken over all $n$-variable LTFs and all $n \ge 0$. Benjamini, Kalai, and Schramm [Publ. Math. Inst. Hautes Etudes Sci., 90 (1999), pp. 5--43] conjectured that the true value of ${\bf W}^{\leq 1}[{\bf LTF}]$ is $2/\pi$. We make progress on this conjecture by proving that ${\bf W}^{\leq 1}[{\bf LTF}] \geq 1/2 + c$ for some absolute constant $c>0$. 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. Let ${\bf W}^{\leq 1}[{...

[1]  Iosif Pinelis,et al.  An asymptotically Gaussian bound on the Rademacher tails , 2010, 1007.2137.

[2]  D. Garling,et al.  Inequalities: A Journey into Linear Analysis , 2007 .

[3]  Rocco A. Servedio,et al.  Every Linear Threshold Function has a Low-Weight Approximator , 2006, 21st Annual IEEE Conference on Computational Complexity (CCC'06).

[4]  R. Latala,et al.  On the best constant in the Khinchin-Kahane inequality , 1994 .

[5]  Rocco A. Servedio,et al.  Nearly Optimal Solutions for the Chow Parameters Problem and Low-Weight Approximation of Halfspaces , 2012, J. ACM.

[6]  I. Pinelis Extremal Probabilistic Problems and Hotelling's $T^2$ Test Under a Symmetry Condition , 1994, math/0701806.

[7]  Feller William,et al.  An Introduction To Probability Theory And Its Applications , 1950 .

[8]  Rocco A. Servedio,et al.  Improved Approximation of Linear Threshold Functions , 2009, 2009 24th Annual IEEE Conference on Computational Complexity.

[9]  Ron Holzman,et al.  On the product of sign vectors and unit vectors , 1992, Comb..

[10]  S. Montgomery-Smith The distribution of Rademacher sums , 1990 .

[11]  Wolfgang Maass,et al.  How fast can a threshold gate learn , 1994, COLT 1994.

[12]  Rocco A. Servedio,et al.  A Robust Khintchine Inequality, and Algorithms for Computing Optimal Constants in Fourier Analysis and High-Dimensional Geometry , 2013, ICALP.

[13]  James Renegar,et al.  A faster PSPACE algorithm for deciding the existential theory of the reals , 1988, [Proceedings 1988] 29th Annual Symposium on Foundations of Computer Science.

[14]  S. Szarek On the best constants in the Khinchin inequality , 1976 .

[15]  Krzysztof Oleszkiewicz,et al.  Comparison of moments via Poincare-type inequality , 1999 .

[16]  Richard K. Guy,et al.  Any Answers Anent These Analytical Enigmas , 1986 .

[17]  Kees Roos,et al.  Robust Solutions of Uncertain Quadratic and Conic-Quadratic Problems , 2002, SIAM J. Optim..

[18]  Rocco A. Servedio,et al.  Testing Halfspaces , 2007, SIAM J. Comput..

[19]  Jeffrey C. Jackson,et al.  Uniform-Distribution Learnability of Noisy Linear Threshold Functions with Restricted Focus of Attention , 2006, COLT.

[20]  Krzysztof Oleszkiewicz On the Stein property of Rademacher sequences , 1996 .

[21]  Nathan Linial,et al.  Spectral properties of threshold functions , 1994, Comb..

[22]  Noga Alon,et al.  Nonnegative k-sums, fractional covers, and probability of small deviations , 2012, J. Comb. Theory, Ser. B.

[23]  Ryan O'Donnell,et al.  The chow parameters problem , 2008, SIAM J. Comput..

[24]  Ryan O'Donnell,et al.  The Chow Parameters Problem , 2011, SIAM J. Comput..

[25]  C. Schütt,et al.  Projection constants of symmetric spaces and variants of Khintchine's inequality , 1999 .

[26]  Shai Ben-David,et al.  Learning with restricted focus of attention , 1993, COLT '93.

[27]  Anthony Man-Cho So,et al.  Improved approximation bound for quadratic optimization problems with orthogonality constraints , 2009, SODA.

[28]  Pawel Hitczenko,et al.  On the Rademacher Series , 1994 .

[29]  Krzysztof Oleszkiewicz,et al.  Comparison of Moments of Sums of Independent Random Variables and Differential Inequalities , 1996 .

[30]  Zhi-Quan Luo,et al.  Semidefinite Relaxation Bounds for Indefinite Homogeneous Quadratic Optimization , 2008, SIAM J. Optim..

[31]  J. Wissel,et al.  On the Best Constants in the Khintchine Inequality , 2007 .

[32]  I. Benjamini,et al.  Noise sensitivity of Boolean functions and applications to percolation , 1998 .

[33]  V. Bentkus On Hoeffding’s inequalities , 2004, math/0410159.

[34]  William Feller,et al.  An Introduction to Probability Theory and Its Applications , 1967 .

[35]  Ryan O'Donnell,et al.  Analysis of Boolean Functions , 2014, ArXiv.

[36]  Y. Peres Noise Stability of Weighted Majority , 2004, math/0412377.

[37]  Rocco A. Servedio,et al.  Bounded Independence Fools Halfspaces , 2009, 2009 50th Annual IEEE Symposium on Foundations of Computer Science.

[38]  Andrew Zimmer,et al.  Hoeffding's Inequality , 2014 .

[39]  V. Bentkus,et al.  A tight Gaussian bound for weighted sums of Rademacher random variables , 2013, 1307.3451.