On some extensions of the FKN theorem
TL;DR: A simple and elementary proof of Friedgut, Kalai, and Naor's result that if Var(jSj) is much smaller than Var(S), then the sum is largely determined by one of the summands is provided.
Abstract: Let S = a1r1+ a2r2+ + anrn be a weighted Rademacher sum. Friedgut, Kalai, and Naor have shown that if Var(jSj) is much smaller than Var(S), then the sum is largely determined by one of the summands. We provide a simple and elementary proof of this result, strengthen it, and extend it in various ways to a more general setting.
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Book•
05 Jun 2014TL;DR: This text gives a thorough overview of Boolean functions, beginning with the most basic definitions and proceeding to advanced topics such as hypercontractivity and isoperimetry, and includes a "highlight application" such as Arrow's theorem from economics.
Abstract: Boolean functions are perhaps the most basic objects of study in theoretical computer science. They also arise in other areas of mathematics, including combinatorics, statistical physics, and mathematical social choice. The field of analysis of Boolean functions seeks to understand them via their Fourier transform and other analytic methods. This text gives a thorough overview of the field, beginning with the most basic definitions and proceeding to advanced topics such as hypercontractivity and isoperimetry. Each chapter includes a "highlight application" such as Arrow's theorem from economics, the Goldreich-Levin algorithm from cryptography/learning theory, Hstad's NP-hardness of approximation results, and "sharp threshold" theorems for random graph properties. The book includes roughly 450 exercises and can be used as the basis of a one-semester graduate course. It should appeal to advanced undergraduates, graduate students, and researchers in computer science theory and related mathematical fields.
867 citations
TL;DR: If a Boolean function f is close to a characteristic function g of a subcube of dimension n-1, then the entropy of Tϵf is at most that of Tϵg.
Abstract: Let $0 be a noise parameter, and let $T_{ \epsilon }$ be the noise operator acting on functions on the Boolean cube $ \{0,1\}^{n}$ . Let $f$ be a nonnegative function on $ \{0,1\}^{n}$ . We upper bound the entropy of $T_{ \epsilon } f$ by the average entropy of conditional expectations of $f$ , given sets of roughly $(1-2 \epsilon )^{\vphantom {R^{l}}2} \cdot n$ variables. In information-theoretic terms, we prove the following strengthening of Mrs. Gerber’s Lemma: let $X$ be a random binary vector of length $n$ , and let $Z$ be a noise vector, corresponding to a binary symmetric channel with crossover probability $ \epsilon $ . Then, setting $v = (1-2 \epsilon )^{2} \cdot n$ , we have (up to lower order terms): $H\big (X \oplus Z\big ) \ge n \cdot H_{2} \big ( \epsilon + (1-2 \epsilon ) \cdot H_{2}^{-1}\big (({ \mathop {\mathbb E}_{|B| = v} H\big (\{X_{i}\}_{i\in B}\big )}/{v})\big )\big )$ . Assuming $ \epsilon \ge 1/2 - \delta $ , for some absolute constant $\delta > 0$ , this inequality, combined with a strong version of a theorem of Friedgut et al. , due to Jendrej et al. , shows that if a Boolean function $f$ is close to a characteristic function $g$ of a subcube of dimension $n-1$ , then the entropy of $T_{ \epsilon } f$ is at most that of $T_{ \epsilon } g$ . Taken together with a recent result of Ordentlich et al. , this shows that the most informative Boolean function conjecture of Courtade and Kumar holds for high noise $ \epsilon \ge 1/2 - \delta $ . Namely, if $X$ is uniformly distributed in $ \{0,1\}^{n}$ and $Y$ is obtained by flipping each coordinate of $X$ independently with probability $ \epsilon $ , then, provided $ \epsilon \ge 1/2 - \delta $ , for any Boolean function $f$ holds $I \left ({ f(X);Y }\right )\le 1 - H( \epsilon )$ .
39 citations
Journal Article•
TL;DR: It is proved that if a Boolean function f is close (in $L^2$-distance) to an affine function $\ell(x_1,...,x_n) = c_0 + \sum_i c_i x_i$, then $f$ is close to a Boolean affinefunction (which necessarily depends on at most one coordinate).
Abstract: The Friedgut–Kalai–Naor theorem states that if a Boolean function f : t0, 1u N t0, 1u is close (in L-distance) to an affine function `px1, . . . , xnq “ c0 ` ř i cixi, then f is close to a Boolean affine function (which necessarily depends on at most one coordinate). We prove a similar theorem for functions defined over `rns k “ tpx1, . . . , xnq P t0, 1u : ř i xi “ ku.
38 citations
29 May 2016
TL;DR: In this paper, the authors show that for one-block decoupled queries, Ck, Dk are significantly better than those known for "full decoupling" when x, y, z are independent Gaussians.
Abstract: Let f(x) = f(x1, ..., xn) = Σ|S|≤k aS IIi∈S xi be an n-variate real multilinear polynomial of degree at most k, where S ⊆ [n] = {1, 2, ..., n}. For its one-block decoupled version.
[EQUATION]
we show tail-bound comparisons of the form
[EQUATION]
Our constants Ck, Dk are significantly better than those known for "full decoupling". For example, when x, y, z are independent Gaussians we obtain Ck = Dk = O(k); when x, y, z are ±1 random variables we obtain Ck = O(k2), Dk = kO(k). By contrast, for full decoupling only Ck = Dk = kO(k) is known in these settings.
We describe consequences of these results for query complexity (related to conjectures of Aaronson and Ambainis) and for analysis of Boolean functions (including an optimal sharpening of the DFKO Inequality).
14 citations
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1,261 citations
"On some extensions of the FKN theor..." refers methods in this paper
...ACM Classification: G.3 AMS Classification: 60E15, 42C10 Key words and phrases: independent random variables, Rademacher variables, absolute value variation, Fourier expansion, Irit Dinur PCP proof...
[...]
Book•
05 Jun 2014TL;DR: This text gives a thorough overview of Boolean functions, beginning with the most basic definitions and proceeding to advanced topics such as hypercontractivity and isoperimetry, and includes a "highlight application" such as Arrow's theorem from economics.
Abstract: Boolean functions are perhaps the most basic objects of study in theoretical computer science. They also arise in other areas of mathematics, including combinatorics, statistical physics, and mathematical social choice. The field of analysis of Boolean functions seeks to understand them via their Fourier transform and other analytic methods. This text gives a thorough overview of the field, beginning with the most basic definitions and proceeding to advanced topics such as hypercontractivity and isoperimetry. Each chapter includes a "highlight application" such as Arrow's theorem from economics, the Goldreich-Levin algorithm from cryptography/learning theory, Hstad's NP-hardness of approximation results, and "sharp threshold" theorems for random graph properties. The book includes roughly 450 exercises and can be used as the basis of a one-semester graduate course. It should appeal to advanced undergraduates, graduate students, and researchers in computer science theory and related mathematical fields.
867 citations
TL;DR: In this paper, the conditions générales d'utilisation (http://www.numdam.org/legal.php) are defined, i.e., the copie ou impression de ce fichier doit contenir la présente mention de copyright.
Abstract: © Annales de l’institut Fourier, 1970, tous droits réservés. L’accès aux archives de la revue « Annales de l’institut Fourier » (http://annalif.ujf-grenoble.fr/) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/legal.php). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.
479 citations
"On some extensions of the FKN theor..." refers methods in this paper
...ACM Classification: G.3 AMS Classification: 60E15, 42C10 Key words and phrases: independent random variables, Rademacher variables, absolute value variation, Fourier expansion, Irit Dinur PCP proof...
[...]