Related papers: An Improved Upper Bound for the Most Informative B…

Let $\lambda$ denote the Liouville function. We show that as $X \rightarrow \infty$, $$ \int_{X}^{2X} \sup_{\alpha} \left | \sum_{x < n \leq x + H} \lambda(n) e(-\alpha n) \right | dx = o ( X H) $$ for all $H \geq X^{\theta}$ with $\theta >…

We study correlation bounds under pairwise independent distributions for functions with no large Fourier coefficients. Functions in which all Fourier coefficients are bounded by $\delta$ are called $\delta$-{\em uniform}. The search for…

One of the major outstanding foundational problems about boolean functions is the sensitivity conjecture, which (in one of its many forms) asserts that the degree of a boolean function (i.e. the minimum degree of a real polynomial that…

We study the probability of Boolean functions with small max influence to become constant under random restrictions. Let $f$ be a Boolean function such that the variance of $f$ is $\Omega(1)$ and all its individual influences are bounded by…

The probabilistic degree of a Boolean function $f:\{0,1\}^n\rightarrow \{0,1\}$ is defined to be the smallest $d$ such that there is a random polynomial $\mathbf{P}$ of degree at most $d$ that agrees with $f$ at each point with high…

We derive a general upper bound to mutual information in terms of the Fisher information. The bound may be further used to derive a lower bound for the Bayesian quadratic cost. These two provide alternatives to other inequalities in the…

We give a combinatorial proof of the result of Kahn, Kalai, and Linial, which states that every balanced boolean function on the $n$-dimensional boolean cube has a variable with influence of at least Omega(\frac{log n}{n}). The methods of…

Algebraic immunity of Boolean function $f$ is defined as the minimal degree of a nonzero $g$ such that $fg=0$ or $(f+1)g=0$. Given a positive even integer $n$, it is found that the weight distribution of any $n$-variable symmetric Boolean…

In this paper we provide new bounds on classical and quantum distributional communication complexity in the two-party, one-way model of communication. In the classical model, our bound extends the well known upper bound of Kremer, Nisan and…

We prove weighted versions of the 2D Restriction Conjecture for the unit sphere in $\mathbb{R}^2$. Our results involve the weight functions $(1+|x|)^\alpha(1+|y|)^\beta$ and $(1+|x|+|y|)^\gamma$ with $\alpha,\beta,\gamma\geq 0$.

For $0 \leq \beta < \alpha < 1$ the distribution $\mathcal{H}$ over Boolean functions $h \colon \{-1, 1\}^d \to \{-1, 1\}$ that minimizes the expression \begin{equation*} \rho_{\alpha, \beta} = \frac{\log(1/\Pr_{\substack{h \sim \mathcal{H}…

We discuss some open problems concerning the maximal spread of coherent distributions. We prove a sharp bound on $\mathbb{E}|X-Y|^{\alpha}$ for $(X,Y)$ coherent and $\alpha \le 2$, and establish a novel connection between coherent…

We develop a new technique for proving concentration inequalities which relate between the variance and influences of Boolean functions. Using this technique, we 1. Settle a conjecture of Talagrand [Tal97] proving that $$\int_{\left\{…

We derive a tight lower bound on equivocation (conditional entropy), or equivalently a tight upper bound on mutual information between a signal variable and channel outputs. The bound is in terms of the joint distribution of the signals and…

A lower bound on the minimum error probability for multihypothesis testing is established. The bound, which is expressed in terms of the cumulative distribution function of the tilted posterior hypothesis distribution given the observation…

In this report, we show that all n-variable Boolean function can be represented as polynomial threshold functions (PTF) with at most $0.75 \times 2^n$ non-zero integer coefficients and give an upper bound on the absolute value of these…

Upper and lower bounds on the error probability of linear codes under maximum-likelihood (ML) decoding are shortly surveyed and applied to ensembles of codes on graphs. For upper bounds, focus is put on Gallager bounding techniques and…

Information bottleneck (IB) is a technique for extracting information in one random variable $X$ that is relevant for predicting another random variable $Y$. IB works by encoding $X$ in a compressed "bottleneck" random variable $M$ from…

The analysis of randomized search heuristics on classes of functions is fundamental for the understanding of the underlying stochastic process and the development of suitable proof techniques. Recently, remarkable progress has been made in…

Let $I, J\subset \mathbb{R}$ be closed intervals, and let $H$ be $C^{3}$ smooth real valued function on $I\times J$ with nonvanishing $H_{x}$ and $H_{y}$. Take any fixed positive numbers $a,b$, and let $d\mu$ be a probability measure with…