Related papers: A Note on the Entropy/Influence Conjecture
Every Boolean function can be uniquely represented as a multilinear polynomial. The entropy and the total influence are two ways to measure the concentration of its Fourier coefficients, namely the monomial coefficients in this…
The Fourier Entropy-Influence (FEI) conjecture of Friedgut and Kalai [FK96] seeks to relate two fundamental measures of Boolean function complexity: it states that $H[f] \leq C Inf[f]$ holds for every Boolean function $f$, where $H[f]$…
The total influence of a function is a central notion in analysis of Boolean functions, and characterizing functions that have small total influence is one of the most fundamental questions associated with it. The KKL theorem and the…
In this paper, we prove that most of the boolean functions, $f : \{-1,1\}^n \rightarrow \{-1,1\}$ satisfy the Fourier Entropy Influence (FEI) Conjecture due to Friedgut and Kalai (Proc. AMS'96). The conjecture says that the Entropy of a…
The Fourier Entropy-Influence (FEI) Conjecture of Friedgut and Kalai states that ${\bf H}[f] \leq C \cdot {\bf I}[f]$ holds for every Boolean function $f$, where ${\bf H}[f]$ denotes the spectral entropy of $f$, ${\bf I}[f]$ is its total…
The Fourier-Entropy Influence (FEI) Conjecture states that for any Boolean function $f:\{+1,-1\}^n \to \{+1,-1\}$, the Fourier entropy of $f$ is at most its influence up to a universal constant factor. While the FEI conjecture has been…
This manuscript includes some classical results we select apart from the new results we've found on the Analysis of Boolean Functions and Fourier-Entropy-Influence conjecture. We try to ensure the self-completeness of this work so that…
Given $f:\{-1, 1\}^n \rightarrow \{-1, 1\}$, define the \emph{spectral distribution} of $f$ to be the distribution on subsets of $[n]$ in which the set $S$ is sampled with probability $\widehat{f}(S)^2$. Then the Fourier Entropy-Influence…
Given a Boolean function $f:\{-1,1\}^n\to \{-1,1\}$, the Fourier distribution assigns probability $\widehat{f}(S)^2$ to $S\subseteq [n]$. The Fourier Entropy-Influence (FEI) conjecture of Friedgut and Kalai asks if there exist a universal…
We describe a new construction of Boolean functions. A specific instance of our construction provides a 30-variable Boolean function having min-entropy/influence ratio to be $128/45 \approx 2.8444$ which is presently the highest known value…
In this paper, we prove that the Fourier entropy of an $n$-dimensional boolean function $f$ can be upper-bounded by $O(I(f)+ \sum\limits_{k\in[n]}I_k(f)\log \frac{1}{I_k(f)})$, where $I(f)$ is its total influence and $I_k(f)$ is the…
We study Boolean functions on the $p$-biased hypercube $(\{0,1\}^n,\mu_p^n)$ through the lens of Fourier (spectral) entropy, i.e. the Shannon entropy of the squared $p$-biased Fourier coefficients. Motivated by recent restriction-based…
In this note we consider Boolean functions defined on the discrete cube equipped with a biased product probability measure. We prove that if the spectrum of such a function is concentrated on the first two Fourier levels, then the function…
The theory of influences in product measures has profound applications in theoretical computer science, combinatorics, and discrete probability. This deep theory is intimately connected to functional inequalities and to the Fourier analysis…
A Boolean function $f:\{0,1\}^n \to \{0,1\}$ is said to be noise sensitive if inserting a small random error in its argument makes the value of the function almost unpredictable. Benjamini, Kalai and Schramm showed that if the sum of…
In a recent paper [PRE 62, 4665 (2000)] (quant-ph/0203102) Manfredi and Feix proposed an alternative definition of quantum entropy based on Wigner phase-space distribution functions and discussed its properties. They proposed also some…
About twenty years ago we wrote a paper, "Boolean Functions whose Fourier Transform is Concentrated on the First Two Levels", \cite{FKN}. In it we offered several proofs of the statement that Boolean functions $f(x_1,x_2,\dots,x_n)$, whose…
We study the problem of estimating a monotone function $f:\{0,1\}^d\to[0,1]$ from noisy observations at uniformly random vertices of the Boolean hypercube. As a measure of complexity for the target~$f$, we use the total $L^1$-influence…
We strengthen Han's Fourier entropy-influence inequality $$ H[\widehat{f}] \leq C_{1}I(f) + C_{2}\sum_{i\in [n]}I_{i}(f)\ln\frac{1}{I_{i}(f)} $$ originally proved for $\{-1,1\}$-valued Boolean functions with $C_{1}=3+2\ln 2$ and $C_{2}=1$.…
Consider a Boolean function f on the n-dimensional hypercube, and a set of variables (indexed by) $S \subset \{1,2,\ldots,n\}.$ The coalition influence of the variables S on a function f is the probability that after a random assignment of…