Related papers: The Average Sensitivity of Bounded-Depth Formulas

We establish a lower bound of $\Omega{(\sqrt{n})}$ on the bounded-error quantum query complexity of read-once Boolean functions, providing evidence for the conjecture that $\Omega(\sqrt{D(f)})$ is a lower bound for all Boolean functions.…

We describe the asymptotic behaviour of the minimal heterogeneous $d$-capacity of a small set, which we assume to be a ball for simplicity, in a fixed bounded open set $\Omega\subseteq \mathbb{R}^d$, with $d\geq2$. Two parameters are…

We establish a generic form of hardness amplification for the approximability of constant-depth Boolean circuits by polynomials. Specifically, we show that if a Boolean circuit cannot be pointwise approximated by low-degree polynomials to…

Koiran showed that if a $n$-variate polynomial of degree $d$ (with $d=n^{O(1)}$) is computed by a circuit of size $s$, then it is also computed by a homogeneous circuit of depth four and of size $2^{O(\sqrt{d}\log(d)\log(s))}$. Using this…

For Boolean functions computed by read-once, depth-$D$ circuits with unbounded fan-in over the de Morgan basis, we present an explicit pseudorandom generator with seed length $\tilde{O}(\log^{D+1} n)$. The previous best seed length known…

Kayal, Saha and Tavenas [Theory of Computing, 2018] showed that for all large enough integers $n$ and $d$ such that $d\geq \omega(\log{n})$, any syntactic depth four circuit of bounded individual degree $\delta = o(d)$ that computes the…

We investigate the randomized decision tree complexity of a specific class of read-once threshold functions. A read-once threshold formula can be defined by a rooted tree, every internal node of which is labeled by a threshold function…

We prove that every bounded function $f:\{-1,1\}^n\to[-1,1]$ of degree at most $d$ can be learned with $L_2$-accuracy $\varepsilon$ and confidence $1-\delta$ from $\log(\tfrac{n}{\delta})\,\varepsilon^{-d-1} C^{d^{3/2}\sqrt{\log d}}$ random…

We study the formula complexity of Iterated Sub-Permutation Matrix Multiplication, the logspace-complete problem of computing the product of $k$ $n$-by-$n$ Boolean matrices with at most a single $1$ in each row and column. For all $d \le…

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…

Let $\Omega$ be homogeneous of degree zero, have vanishing moment of order one on the unit sphere $\mathbb {S}^{d-1}$($d\ge 2$). In this paper, our object of investigation is the following rough non-standard singular integral operator…

We show that any depth-$d$ circuit for determining whether an $n$-node graph has an $s$-to-$t$ path of length at most $k$ must have size $n^{\Omega(k^{1/d}/d)}$. The previous best circuit size lower bounds for this problem were…

For estimating a positive normal mean, Zhang and Woodroofe (2003) as well as Roe and Woodroofe (2000) investigate 100($1-\alpha)%$ HPD credible sets associated with priors obtained as the truncation of noninformative priors onto the…

Consider the finite regular language L_n = {w0 : w \in {0,1}^*, |w| \le n}. It was shown by Ambainis, Nayak, Ta-Shma and Vazirani that while this language is accepted by a deterministic finite automaton of size O(n), any one-way quantum…

We study the formula complexity of the word problem $\mathsf{Word}_{S_n,k} : \{0,1\}^{kn^2} \to \{0,1\}$: given $n$-by-$n$ permutation matrices $M_1,\dots,M_k$, compute the $(1,1)$-entry of the matrix product $M_1\cdots M_k$. An important…

In 2009, Roeglin and Teng showed that the smoothed number of Pareto optimal solutions of linear multi-criteria optimization problems is polynomially bounded in the number $n$ of variables and the maximum density $\phi$ of the semi-random…

We study gapped scale-sensitive dimensions of a function class in both sequential and non-sequential settings. We demonstrate that covering numbers for any uniformly bounded class are controlled above by these gapped dimensions,…

Motivated by applications to the study of depth functions for tree-indexed random variables generated by point processes, we describe functional limit theorems for the intensity measure of point processes. Specifically, we establish uniform…

We give the first super-polynomial separation in the power of bounded-depth boolean formulas vs. circuits. Specifically, we consider the problem Distance $k(n)$ Connectivity, which asks whether two specified nodes in a graph of size $n$ are…

In this paper we research a model of multilayer circuits with a single logical layer. We consider $\lambda$-separable graphs as a support for circuits. We establish the Shannon function lower bound $\max \bigl(\frac{2^n}{n}, \frac{2^n (1 -…