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Register a new userBoolean functions whose Fourier transform is concentrated on pairwise disjoint subsets of the input
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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.
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We initiate the study of Boolean function analysis on high-dimensional expanders. We give a random-walk based definition of high dimensional expansion, which coincides with the earlier definition in terms of two-sided link expanders. Using this definition, we describe an analogue of the Fourier expansion and the Fourier levels of the Boolean hypercube for simplicial complexes. Our analogue is a decomposition into approximate eigenspaces of random walks associated with the simplicial complexes. We then use this decomposition to extend the Friedgut-Kalai-Naor theorem to high-dimensional expanders. Our results demonstrate that a high-dimensional expander can sometimes serve as a sparse model for the Boolean slice or hypercube, and quite possibly additional results from Boolean function analysis can be carried over to this sparse model. Therefore, this model can be viewed as a derandomization of the Boolean slice, containing only $|X(k-1)|=O(n)$ points in contrast to $binom{n}{k}$ points in the $(k)$-slice (which consists of all $n$-bit strings with exactly $k$ ones). Our random-walk definition and the decomposition has the additional advantage that they extend to the more general setting of posets, which include both high-dimensional expanders and the Grassmann poset, which appears in recent works on the unique games conjecture.
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 $p_n$.
We study parity decision trees for Boolean functions. The motivation of our study is the log-rank conjecture for XOR functions and its connection to Fourier analysis and parity decision tree complexity. Let f be a Boolean function with Fourier support S and Fourier sparsity k. 1) We prove via the probabilistic method that there exists a parity decision tree of depth O(sqrt k) that computes f. This matches the best known upper bound on the parity decision tree complexity of Boolean functions (Tsang, Wong, Xie, and Zhang, FOCS 2013). Moreover, while previous constructions (Tsang et al., FOCS 2013, Shpilka, Tal, and Volk, Comput. Complex. 2017) build the trees by carefully choosing the parities to be queried in each step, our proof shows that a naive sampling of the parities suffices. 2) We generalize the above result by showing that if the Fourier spectra of Boolean functions satisfy a natural folding property, then the above proof can be adapted to establish existence of a tree of complexity polynomially smaller than O(sqrt k). We make a conjecture in this regard which, if true, implies that the communication complexity of an XOR function is bounded above by the fourth root of the rank of its communication matrix, improving upon the previously known upper bound of square root of rank (Tsang et al., FOCS 2013, Lovett, J. ACM. 2016). 3) It can be shown by elementary techniques that for any Boolean function f and all pairs (alpha, beta) of parities in S, there exists another pair (gamma, delta) of parities in S such that alpha + beta = gamma + delta. We show, among other results, that there must exist several gamma in F_2^n such that there are at least three pairs (alpha_1, alpha_2) of parities in S with alpha_1 + alpha_2 = gamma.
We examine a new path transform on 1-dimensional simple random walks and Brownian motion, the quantile transform. This transformation relates to identities in fluctuation theory due to Wendel, Port, Dassios and others, and to discrete and Browni
Let $mathcal{H}$ denote a collection of subsets of ${1,2,ldots,n}$, and assign independent random variables uniformly distributed over $[0,1]$ to the $n$ elements. Declare an element $p$-present if its corresponding value is at most $p$. In this paper, we quantify how much the observation of the $r$-present ($r>p$) set of elements affects the probability that the set of $p$-present elements is contained in $mathcal{H}$. In the context of percolation, we find that this question is closely linked to the near-critical regime. As a consequence, we show that for every $r>1/2$, bond percolation on the subgraph of the square lattice given by the set of $r$-present edges is almost surely noise sensitive at criticality, thus generalizing a result due to Benjamini, Kalai and Schramm.
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