Related papers: Polynomials, Quantum Query Complexity, and Grothen…
It has long been known that any Boolean function that depends on n input variables has both degree and exact quantum query complexity of Omega(log n), and that this bound is achieved for some functions. In this paper we study the case of…
Polynomial approximations to boolean functions have led to many positive results in computer science. In particular, polynomial approximations to the sign function underly algorithms for agnostically learning halfspaces, as well as…
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…
We prove various theorems on approximation using polynomials with integer coefficients in the Bernstein basis of any given order. In the extreme, we draw the coefficients from $\{ \pm 1\}$ only. A basic case of our results states that for…
We study approximation of Boolean functions by low-degree polynomials over the ring $\mathbb{Z}/2^k\mathbb{Z}$. More precisely, given a Boolean function $F:\{0,1\}^n \rightarrow \{0,1\}$, define its $k$-lift to be $F_k:\{0,1\}^n \rightarrow…
We study the computation complexity of Boolean functions in the quantum black box model. In this model our task is to compute a function $f:\{0,1\}\to\{0,1\}$ on an input $x\in\{0,1\}^n$ that can be accessed by querying the black box.…
The $\epsilon$-approximate degree $deg_\epsilon(f)$ of a Boolean function $f$ is the least degree of a real-valued polynomial that approximates $f$ pointwise to error $\epsilon$. The approximate degree of $f$ is at least $k$ iff there…
We give two approximation algorithms solving the Stochastic Boolean Function Evaluation (SBFE) problem for symmetric Boolean functions. The first is an $O(\log n)$-approximation algorithm, based on the submodular goal-value approach of…
In this work, we consider a new type of Fourier-like representation of Boolean function $f\colon\{+1,-1\}^n\to\{+1,-1\}$ \[ f(x) = \cos\left(\pi\sum_{S\subseteq[n]}\phi_S \prod_{i\in S} x_i\right). \] This representation, which we call the…
We give a new presentation of the main result of Arunachalam, Bri\"et and Palazuelos (SICOMP'19) and show that quantum query algorithms are characterized by a new class of polynomials which we call Fourier completely bounded polynomials. We…
Let X = (x_0,...,x_{n-1})$ be a sequence of n numbers. For \epsilon > 0, we say that x_i is an \epsilon-approximate median if the number of elements strictly less than x_i, and the number of elements strictly greater than x_i are each less…
We make the case for neural network objects and extend an already existing neural network calculus explained in detail in Chapter 2 on \cite{bigbook}. Our aim will be to show that, yes, indeed, it makes sense to talk about neural network…
In this note, we develop a bounded-error quantum algorithm that makes $\tilde O(n^{1/4}\varepsilon^{-1/2})$ queries to a Boolean function $f$, accepts a monotone function, and rejects a function that is $\varepsilon$-far from being…
The vectorial Boolean functions are employed in cryptography to build block coding algorithms. An important criterion on these functions is their resistance to the differential cryptanalysis. Nyberg defined the notion of almost perfect…
Let $g: \{-1,1\}^k \to \{-1,1\}$ be any Boolean function and $q_1,\dots,q_k$ be any degree-2 polynomials over $\{-1,1\}^n.$ We give a \emph{deterministic} algorithm which, given as input explicit descriptions of $g,q_1,\dots,q_k$ and an…
Polynomial representations of Boolean functions over various rings such as $\mathbb{Z}$ and $\mathbb{Z}_m$ have been studied since Minsky and Papert (1969). From then on, they have been employed in a large variety of fields including…
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…
We give a new framework for proving the existence of low-degree, polynomial approximators for Boolean functions with respect to broad classes of non-product distributions. Our proofs use techniques related to the classical moment problem…
The relationship between quantum physics and discrete mathematics is reviewed in this article. The Boolean functions unitary representation is considered. The relationship between Zhegalkin polynomial, which defines the algebraic normal…
A theorem is proved concerning approximation of analytic functions by multivariate polynomials in the $s$-dimensional hypercube. The geometric convergence rate is determined not by the usual notion of degree of a multivariate polynomial,…