Related papers: A Differential Equation Approach to the Most-Infor…
We prove the "Most informative boolean function" conjecture of Courtade and Kumar for high noise $\epsilon \ge 1/2 - \delta$, for some absolute constant $\delta > 0$. Namely, if $X$ is uniformly distributed in $\{0,1\}^n$ and $Y$ is…
We introduce a simply stated conjecture regarding the maximum mutual information a Boolean function can reveal about noisy inputs. Specifically, let $X^n$ be i.i.d. Bernoulli(1/2), and let $Y^n$ be the result of passing $X^n$ through a…
Suppose $X$ is a uniformly distributed $n$-dimensional binary vector and $Y$ is obtained by passing $X$ through a binary symmetric channel with crossover probability $\alpha$. A recent conjecture by Courtade and Kumar postulates that…
We prove the Courtade-Kumar conjecture, for several classes of n-dimensional Boolean functions, for all $n \geq 2$ and for all values of the error probability of the binary symmetric channel, $0 \leq p \leq 1/2$. This conjecture states that…
Suppose $\XX{N}$ is a uniformly distributed $N$-dimensional binary vector and $\YY{N}$ is obtained by passing $\XX{N}$ through a binary symmetric channel with crossover probability $\alpha$. Recently, Courtade and Kumar postulates that…
We prove the Courtade-Kumar conjecture, which states that the mutual information between any Boolean function of an $n$-dimensional vector of independent and identically distributed inputs to a memoryless binary symmetric channel and the…
This paper develops upper and lower bounds for the probability of Boolean functions by treating multiple occurrences of variables as independent and assigning them new individual probabilities. We call this approach dissociation and give an…
We prove the Courtade-Kumar conjecture, for certain classes of $n$-dimensional Boolean functions, $\forall n\geq 2$ and for all values of the error probability of the binary symmetric channel, $\forall 0 \leq p \leq \frac{1}{2}$. Let…
In this work, we consider the problem of bounding the values of a covariance function corresponding to a continuous-time stationary stochastic process or signal. Specifically, for two signals whose covariance functions agree on a finite…
This paper investigates the learnability of the nonlinearity property of Boolean functions using neural networks. We train encoder style deep neural networks to learn to predict the nonlinearity of Boolean functions from examples of…
Monotone Boolean functions are a structurally important class of Boolean functions, but their restricted form imposes strong limitations on achievable nonlinearity. In this paper, we investigate whether evolutionary computation can evolve…
Boolean functions are mathematical objects used in diverse applications. Different applications also have different requirements, making the research on Boolean functions very active. In the last 30 years, evolutionary algorithms have been…
Building on the recently introduced notion of Boolean entropy, we define the corresponding Boolean Fisher information via a de Bruijn identity. We study the monotonicity of this Fisher information in the Boolean Central Limit Theorem and…
A linear inference is a valid inequality of Boolean algebra in which each variable occurs at most once on each side. In this work we leverage recently developed graphical representations of linear formulae to build an implementation that is…
We study the extremal competitive ratio of Boolean function evaluation. We provide the first non-trivial lower and upper bounds for classes of Boolean functions which are not included in the class of monotone Boolean functions. For the…
This work presents a study of perturbations of symmetric Boolean functions. In particular, it establishes a connection between exponential sums of these perturbations and Diophantine equations of the form $$ \sum_{l=0}^n \binom{n}{l}…
We prove two conjectures on correlation inequalities for functions that are linear combinations of unimodal Boolean monotone nondecreasing functions
Let $X^n$ be a uniformly distributed $n$-dimensional binary vector, and $Y^n$ be the result of passing $X^n$ through a binary symmetric channel (BSC) with crossover probability $\alpha$. A recent conjecture postulated by Courtade and Kumar…
Determining functionals are tools to describe the finite dimensional long-term dynamics of infinite dimensional dynamical systems. There also exist several applications to infinite dimensional {\em random} dynamical systems. In these…
This paper presents a method to detect and recognize symmetries in Boolean functions. The idea is to use information theoretic measures of Boolean functions to detect sub-space of possible symmetric variables. Coupled with the new…