Related papers: Strong noise sensitivity and random graphs
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…
A first step is taken towards understanding often observed non-robustness phenomena of deep neural net (DNN) classifiers. This is done from the perspective of Boolean functions by asking if certain sequences of Boolean functions represented…
It is shown that a large class of events in a product probability space are highly sensitive to noise, in the sense that with high probability, the configuration with an arbitrary small percent of random errors gives almost no prediction…
In this paper we study noise sensitivity and threshold phenomena for Poisson Voronoi percolation on $\mathbb{R}^2$. In the setting of Boolean functions, both threshold phenomena and noise sensitivity can be understood via the study of…
The study of noise sensitivity of Boolean functions was initiated in a seminal paper of Benjamini, Kalai and Schramm, published in 1999. While this study has revealed fascinating phenomena in the context of Bernoulli percolation, few…
We study noise sensitivity of properties of the largest components $({\cal C}_j)_{j\geq 1}$ of the random graph ${\cal G}(n,p)$ in its critical window $p=(1+\lambda n^{-1/3})/n$. For instance, is the property "$|{\cal C}_1|$ exceeds its…
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 study functions on the infinite-dimensional Hamming cube $\{-1,1\}^\infty$, in particular Boolean functions into $\{-1,1\}$, generalising results on analysis of Boolean functions on $\{-1,1\}^n$ for $n\in\mathbb{N}$. The notion of noise…
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…
We investigate the effect of noise on Random Boolean Networks. Noise is implemented as a probability $p$ that a node does not obey its deterministic update rule. We define two order parameters, the long-time average of the Hamming distance…
A natural measure of smoothness of a Boolean function is its sensitivity (the largest number of Hamming neighbors of a point which differ from it in function value). The structure of smooth or equivalently low-sensitivity functions is still…
We present a regularity lemma for Boolean functions $f:\{-1,1\}^n \to \{-1,1\}$ based on noisy influence, a measure of how locally correlated $f$ is with each input bit. We provide an application of the regularity lemma to weaken the…
Recently the study of noise sensitivity and noise stability of Boolean functions has received considerable attention. The purpose of this paper is to extend these notions in a natural way to a different class of perturbations, namely those…
Assign independent weights to the edges of the square lattice, from the uniform distribution on $\{a,b\}$ for some $0<a<b<\infty$. The weighted graph induces a random metric on $\mathbb{Z}^2$. Let $T_n$ denote the distance between $(0,0)$…
Keller and Kindler recently established a quantitative version of the famous Benjamini~--Kalai--Schramm Theorem on noise sensitivity of Boolean functions. The result was extended to the continuous Gaussian setting by Keller, Mossel and Sen…
We define and study the complexity of robust polynomials for Boolean functions and the related fault-tolerant quantum decision trees, where input bits are perturbed by noise. We compare several different possible definitions. Our main…
This paper considers the problem of approximating a Boolean function $f$ using another Boolean function from a specified class. Two classes of approximating functions are considered: $k$-juntas, and linear Boolean functions. The $n$ input…
The dichotomy between noise-stable and (completely) noise-sensitive stochastic models is of recent interest in probability theory. Of particular interest is the study of lattice models coming from statistical physics. The Fourier transform…
Gaussian bounds on noise correlation of functions play an important role in hardness of approximation, in quantitative social choice theory and in testing. The author (2008) obtained sharp gaussian bounds for the expected correlation of…
The sensitivity of a Boolean function f is the maximum over all inputs x, of the number of sensitive coordinates of x. The well-known sensitivity conjecture of Nisan (see also Nisan and Szegedy) states that every sensitivity-s Boolean…