Related papers: The "Most informative boolean function" conjecture…
The Komlos conjecture in discrepancy theory states that for some constant K and for any m by n matrix A whose columns lie in the unit ball there exists a +/- 1 vector x such that the infinity norm of Ax is bounded above by K. This…
We define tests of boolean functions which distinguish between linear (or quadratic) polynomials, and functions which are very far, in an appropriate sense, from these polynomials. The tests have optimal or nearly optimal trade-offs between…
We characterize the power of constant-depth Boolean circuits in generating uniform symmetric distributions. Let $f\colon\{0,1\}^m\to\{0,1\}^n$ be a Boolean function where each output bit of $f$ depends only on $O(1)$ input bits. Assume the…
One of the most controversial problems in neural decoding is quantifying the information loss caused by ignoring noise correlations during optimal brain computations. For more than a decade, the measure here called $ \Delta I^{DL} $ has…
A function $f:\ \{-1,1\}^n\rightarrow \mathbb{R}$ is called pseudo-Boolean. It is well-known that each pseudo-Boolean function $f$ can be written as $f(x)=\sum_{I\in {\cal F}}\hat{f}(I)\chi_I(x),$ where ${\cal F}\subseteq \{I:\ I\subseteq…
Error-correcting codes are efficient methods for handling \emph{noisy} communication channels in the context of technological networks. However, such elaborate methods differ a lot from the unsophisticated way biological entities are…
Consider informative selection of a sample from a finite population. Responses are realized as independent and identically distributed (i.i.d.) random variables with a probability density function (p.d.f.) f, referred to as the…
How should one combine noisy information from diverse sources to make an inference about an objective ground truth? This frequently recurring, normative question lies at the core of statistics, machine learning, policy-making, and everyday…
We find estimates on the norms commutators of the form [f(x), y] in terms of the norm of [x, y] assuming that x and y are contractions in a C*-algebra A, with x normal and with spectrum within the domain of f. In particular we discuss [x^2,…
We prove a lower bound on the maximum of the Riemann zeta function in a typical short interval on the critical line. Together with the upper bound from the previous work of the authors, this implies tightness of $$ \max_{|h|\leq…
This paper considers estimation of a quantized constant in noise when using uniform and nonuniform quantizers. Estimators based on simple arithmetic averages, on sample statistical moments and on the maximum-likelihood procedure are…
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…
We study the fundamental problem of learning the parameters of a high-dimensional Gaussian in the presence of noise -- where an $\varepsilon$-fraction of our samples were chosen by an adversary. We give robust estimators that achieve…
We introduce a theoretical model of information acquisition under resource limitations in a noisy environment. An agent must guess the truth value of a given Boolean formula $\varphi$ after performing a bounded number of noisy tests of the…
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…
Interactive coding allows two parties to conduct a distributed computation despite noise corrupting a certain fraction of their communication. Dani et al.\@ (Inf.\@ and Comp., 2018) suggested a novel setting in which the amount of noise is…
Particle physics experiments such as those run in the Large Hadron Collider result in huge quantities of data, which are boiled down to a few numbers from which it is hoped that a signal will be detected. We discuss a simple probability…
In a recent paper by Jonasson and Steif, definitions to describe the volatility of sequences of Boolean functions, \( f_n \colon \{ -1,1 \}^n \to \{ -1,1 \} \) were introduced. We continue their study of how these definitions relate to…
This dissertation shows that careful injection of noise into sample data can substantially speed up Expectation-Maximization algorithms. Expectation-Maximization algorithms are a class of iterative algorithms for extracting maximum…
The HRT (Heil-Ramanathan-Topiwala) conjecture asks whether a finite collection of time-frequency shifts of a non-zero square integrable function on $\mathbb{R}$ is linearly independent. This longstanding conjecture remains largely open even…