Related papers: Sharp Thresholds for Monotone Non Boolean Function…
The hypercontractive inequality on the discrete cube plays a crucial role in many fundamental results in the Analysis of Boolean functions, such as the KKL theorem, Friedgut's junta theorem and the invariance principle. In these results the…
We study the computational power of polynomial threshold functions, that is, threshold functions of real polynomials over the boolean cube. We provide two new results bounding the computational power of this model. Our first result shows…
In a recent result, Knop, Lovett, McGuire and Yuan (STOC 2021) proved the log-rank conjecture for communication complexity, up to log n factor, for any Boolean function composed with AND function as the inner gadget. One of the main tools…
Empirical likelihood is an attractive inferential framework that respects natural parameter boundaries, but existing approaches typically require smoothness of the functional and miscalibrate substantially when these assumptions are…
Items in a test are often used as a basis for making decisions and such tests are therefore required to have good psychometric properties, like unidimensionality. In many cases the sum score is used in combination with a threshold to decide…
Nisan and Szegedy (CC 1994) showed that any Boolean function $f:\{0,1\}^n\rightarrow \{0,1\}$ that depends on all its input variables, when represented as a real-valued multivariate polynomial $P(x_1,\ldots,x_n)$, has degree at least $\log…
Monotone inclusions have a wide range of applications, including minimization, saddle-point, and equilibria problems. We introduce new stochastic algorithms, with or without variance reduction, to estimate a root of the expectation of…
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…
We will present a new method, which enables us to find threshold functions for many properties in random intersection graphs. This method will be used to establish sharp threshold functions in random intersection graphs for k-connectivity,…
The data functions that are studied in the course of functional data analysis are assembled from discrete data, and the level of smoothing that is used is generally that which is appropriate for accurate approximation of the conceptually…
The noise sensitivity of a Boolean function $f: \{0,1\}^n \rightarrow \{0,1\}$ is one of its fundamental properties. A function of a positive noise parameter $\delta$, it is denoted as $NS_{\delta}[f]$. Here we study the algorithmic problem…
Let $\mathcal{F}_{n}^*$ be the set of Boolean functions depending on all $n$ variables. We prove that for any $f\in \mathcal{F}_{n}^*$, $f|_{x_i=0}$ or $f|_{x_i=1}$ depends on the remaining $n-1$ variables, for some variable $x_i$. This…
We give the first non-trivial upper bounds on the average sensitivity and noise sensitivity of polynomial threshold functions. More specifically, for a Boolean function f on n variables equal to the sign of a real, multivariate polynomial…
We prove that for any decision tree calculating a boolean function $f:\{-1,1\}^n\to\{-1,1\}$, \[ \Var[f] \le \sum_{i=1}^n \delta_i \Inf_i(f), \] where $\delta_i$ is the probability that the $i$th input variable is read and $\Inf_i(f)$ is…
We show that if $f\colon S_n \to \{0,1\}$ is $\epsilon$-close to linear in $L_2$ and $\mathbb{E}[f] \leq 1/2$ then $f$ is $O(\epsilon)$-close to a union of "mostly disjoint" cosets, and moreover this is sharp: any such union is close to…
We give a $2^{\tilde{O}(\sqrt{n}/\epsilon)}$-time algorithm for properly learning monotone Boolean functions under the uniform distribution over $\{0,1\}^n$. Our algorithm is robust to adversarial label noise and has a running time nearly…
We address the problem of finding optimal strategies for computing Boolean symmetric functions. We consider a collocated network, where each node's transmissions can be heard by every other node. Each node has a Boolean measurement and we…
We exploit a result by Nerman which shows that conditional limit theorems hold when a certain monotonicity condition is satisfied. Our main result is an application to vertex degrees in random graphs, where we obtain asymptotic normality…
A boolean function $f(x_1,...,x_n)$ is \textit{weakly symmetric} if it is invariant under a transitive permutation group on its variables. A boolean function $f(x_1,...,x_n)$ is \textit{elusive} if we have to check all $x_1$,..., $x_n$ to…
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…