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Register a new userVolatility of Boolean functions
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Johan Jonasson Jeffrey E. Steif
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We study the volatility of the output of a Boolean function when the input bits undergo a natural dynamics. For $n = 1,2,ldots$, let $f_n:{0,1}^{m_n} ra {0,1}$ be a Boolean function and $X^{(n)}(t)=(X_1(t),ldots,X_{m_n}(t))_{t in [0,infty)}$ be a vector of i.i.d. stationary continuous time Markov chains on ${0,1}$ that jump from $0$ to $1$ with rate $p_n in [0,1]$ and from $1$ to $0$ with rate $q_n=1-p_n$. Our object of study will be $C_n$ which is the number of state changes of $f_n(X^{(n)}(t))$ as a function of $t$ during $[0,1]$. We say that the family ${f_n}_{nge 1}$ is volatile if $C_n ra iy$ in distribution as $ntoinfty$ and say that ${f_n}_{nge 1}$ is tame if ${C_n}_{nge 1}$ is tight. We study these concepts in and of themselves as well as investigate their relationship with the recent notions of noise sensitivity and noise stability. In addition, we study the question of lameness which means that $Pro(C_n =0)ra 1$ as $ntoinfty$. Finally, we investigate these properties for a number of standard Boolean functions such as the majority function, iterated 3-majority, the AND/OR function on the binary tree and percolation on certain trees at various levels of the parameter $p_n$.
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Let $T_{epsilon}$ be the noise operator acting on Boolean functions $f:{0, 1}^nto {0, 1}$, where $epsilonin[0, 1/2]$ is the noise parameter. Given $alpha>1$ and fixed mean $mathbb{E} f$, which Boolean function $f$ has the largest $alpha$-th moment $mathbb{E}(T_epsilon f)^alpha$? This question has close connections with noise stability of Boolean functions, the problem of non-interactive correlation distillation, and Courtade-Kumars conjecture on the most informative Boolean function. In this paper, we characterize maximizers in some extremal settings, such as low noise ($epsilon=epsilon(n)$ is close to 0), high noise ($epsilon=epsilon(n)$ is close to 1/2), as well as when $alpha=alpha(n)$ is large. Analogous results are also established in more general contexts, such as Boolean functions defined on discrete torus $(mathbb{Z}/pmathbb{Z})^n$ and the problem of noise stability in a tree model.
The subject of this textbook is the analysis of Boolean functions. Roughly speaking, this refers to studying Boolean functions $f : {0,1}^n to {0,1}$ via their Fourier expansion and other analytic means. Boolean functions are perhaps the most basic object of study in theoretical computer science, and Fourier analysis has become an indispensable tool in the field. The topic has also played a key role in several other areas of mathematics, from combinatorics, random graph theory, and statistical physics, to Gaussian geometry, metric/Banach spaces, and social choice theory. The intent of this book is both to develop the foundations of the field and to give a wide (though far from exhaustive) overview of its applications. Each chapter ends with a highlight showing the power of analysis of Boolean functions in different subject areas: property testing, social choice, cryptography, circuit complexity, learning theory, pseudorandomness, hardness of approximation, concrete complexity, and random graph theory. The book can be used as a reference for working researchers or as the basis of a one-semester graduate-level course. The author has twice taught such a course at Carnegie Mellon University, attended mainly by graduate students in computer science and mathematics but also by advanced undergraduates, postdocs, and researchers in adjacent fields. In both years most of Chapters 1-5 and 7 were covered, along with parts of Chapters 6, 8, 9, and 11, and some additional material on additive combinatorics. Nearly 500 exercises are provided at the ends of the books chapters.
We consider Boolean functions f:{-1,1}^n->{-1,1} that are close to a sum of independent functions on mutually exclusive subsets of the variables. We prove that any such function is close to just a single function on a single subset. We also consider Boolean functions f:R^n->{-1,1} that are close, with respect to any product distribution over R^n, to a sum of their variables. We prove that any such function is close to one of the variables. Both our results are independent of the number of variables, but depend on the variance of f. I.e., if f is epsilon*Var(f)-close to a sum of independent functions or random variables, then it is O(epsilon)-close to one of the independent functions or random variables, respectively. We prove that this dependence on Var(f) is tight. Our results are a generalization of the Friedgut-Kalai-Naor Theorem [FKN02], which holds for functions f:{-1,1}^n->{-1,1} that are close to a linear combination of uniformly distributed Boolean variables.
The goal in the area of functions property testing is to determine whether a given black-box Boolean function has a particular given property or is $varepsilon$-far from having that property. We investigate here several types of properties testing for Boolean functions (identity, correlations and balancedness) using the Deutsch-Jozsa algorithm (for the Deutsch-Jozsa (D-J) problem) and also the amplitude amplification technique. At first, we study here a particular testing problem: namely whether a given Boolean function $f$, of $n$ variables, is identical with a given function $g$ or is $varepsilon$-far from $g$, where $varepsilon$ is the parameter. We present a one-sided error quantum algorithm to deal with this problem that has the query complexity $O(frac{1}{sqrt{varepsilon}})$. Moreover, we show that our quantum algorithm is optimal. Afterwards we show that the classical randomized query complexity of this problem is $Theta(frac{1}{varepsilon})$. Secondly, we consider the D-J problem from the perspective of functional correlations and let $C(f,g)$ denote the correlation of $f$ and $g$. We propose an exact quantum algorithm for making distinction between $|C(f,g)|=varepsilon$ and $|C(f,g)|=1$ using six queries, while the classical deterministic query complexity for this problem is $Theta(2^{n})$ queries. Finally, we propose a one-sided error quantum query algorithm for testing whether one Boolean function is balanced versus $varepsilon$-far balanced using $O(frac{1}{varepsilon})$ queries. We also prove here that our quantum algorithm for balancedness testing is optimal. At the same time, for this balancedness testing problem we present a classical randomized algorithm with query complexity of $O(1/varepsilon^{2})$. Also this randomized algorithm is optimal. Besides, we link the problems considered here together and generalize them to the general case.
We study the asymptotic behaviour of a class of small-noise diffusions driven by fractional Brownian motion, with random starting points. Different scalings allow for different asymptotic properties of the process (small-time and tail behaviours in particular). In order to do so, we extend some results on sample path large deviations for such diffusions. As an application, we show how these results characterise the small-time and tail estimates of the implied volatility for rough volatility models, recently proposed in mathematical finance.
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