Related papers: $k$-Independent Gaussians Fool Polynomial Threshol…

Let x be a random vector coming from any k-wise independent distribution over {-1,1}^n. For an n-variate degree-2 polynomial p, we prove that E[sgn(p(x))] is determined up to an additive epsilon for k = poly(1/epsilon). This answers an open…

We develop a pseudo-random generator to fool degree-$d$ polynomial threshold functions with respect to the Gaussian distribution. For $c>0$ any constant, we construct a pseudo-random generator that fools such functions to within $\epsilon$…

We develop a pseudorandom generator that fools degree-$d$ polynomial threshold functions in $n$ variables with respect to the Gaussian distribution and has seed length $O_{c,d}(\log(n) \epsilon^{-c})$.

Developing explicit pseudorandom generators (PRGs) for prominent categories of Boolean functions is a key focus in computational complexity theory. In this paper, we investigate the PRGs against the functions of degree-$d$ polynomial…

We give a pseudorandom generator that fools degree-$d$ polynomial threshold functions over $n$-dimensional Gaussian space with seed length $\mathrm{poly}(d)\cdot \log n$. All previous generators had a seed length with at least a $2^d$…

We prove a structural result for degree-$d$ polynomials. In particular, we show that any degree-$d$ polynomial, $p$ can be approximated by another polynomial, $p_0$, which can be decomposed as some function of polynomials $q_1,...,q_m$ with…

We study the computational power of polynomial threshold functions, that is, threshold functions of real polynomials over the boolean cube. We provide two new results bounding the computational power of this model. Our first result shows…

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 provide asymptotically sharp bounds for the Gaussian surface area and the Gaussian noise sensitivity of polynomial threshold functions. In particular we show that if $f$ is a degree-$d$ polynomial threshold function, then its Gaussian…

We prove a new bound on the average sensitivity of polynomial threshold functions. In particular we show that a polynomial threshold function of degree $d$ in at most $n$ variables has average sensitivity at most…

A simple way to generate a Boolean function is to take the sign of a real polynomial in $n$ variables. Such Boolean functions are called polynomial threshold functions. How many low-degree polynomial threshold functions are there? The…

We study whether a uniformly random Boolean function $f : \{-1,1\}^p \to \{-1,1\}$ is determined by its Walsh--Fourier coefficients of degree at most $d$. We show that the threshold lies at $p/2$ up to an $O(\sqrt{p \log p})$ window: if \[…

We show that any distribution on {-1,1}^n that is k-wise independent fools any halfspace h with error \eps for k = O(\log^2(1/\eps) /\eps^2). Up to logarithmic factors, our result matches a lower bound by Benjamini, Gurel-Gurevich, and…

A polynomial threshold function (PTF) $f:\mathbb{R}^n \rightarrow \mathbb{R}$ is a function of the form $f(x) = \mathsf{sign}(p(x))$ where $p$ is a polynomial of degree at most $d$. PTFs are a classical and well-studied complexity class…

New sequences of orthogonal polynomials with respect to the weight functions $e^{-x} \rho_\nu(x),\ e^{- 1/x} x^{-1} \rho_{\nu} (x), \rho_{\nu}(x)= 2 x^{\nu/2} K_\nu(2\sqrt x),\ x >0, \nu \in \mathbb{R}$, where $K_\nu(z)$ is the modified…

For $S \subseteq \{0,1\}^n$ a Boolean function $f \colon S \to \{-1,1\}$ is a polynomial threshold function (PTF) of degree $d$ and weight $W$ if there is a polynomial $p$ with integer coefficients of degree $d$ and with sum of absolute…

Let $2^{[n]}$ denote the power set of $[n]:=\{1,2,..., n\}$. A collection $\B\subset 2^{[n]}$ forms a $d$-dimensional {\em Boolean algebra} if there exist pairwise disjoint sets $X_0, X_1,..., X_d \subseteq [n]$, all non-empty with perhaps…

We construct pseudorandom generators that fool functions of halfspaces (threshold functions) under a very broad class of product distributions. This class includes not only familiar cases such as the uniform distribution on the discrete…

A conjecture of Odoni stated over Hilbertian fields $K$ of characteristic zero asserts that for every positive integer $d$, there exists a polynomial $f\in K[x]$ of degree $d$ such that for every positive integer $n$, each iterate $f^{\circ…

We give the first non-trivial upper bounds on the average sensitivity and noise sensitivity of degree-$d$ polynomial threshold functions (PTFs). These bounds hold both for PTFs over the Boolean hypercube and for PTFs over $\R^n$ under the…