Pseudorandom generators for polynomial threshold functions
Proceedings of the forty-second ACM symposium on Theory of computing
Bounding the average sensitivity and noise sensitivity of polynomial threshold functions
Proceedings of the forty-second ACM symposium on Theory of computing
On the exact space complexity of sketching and streaming small norms
SODA '10 Proceedings of the twenty-first annual ACM-SIAM symposium on Discrete Algorithms
Bounded Independence Fools Halfspaces
SIAM Journal on Computing
A PRG for lipschitz functions of polynomials with applications to sparsest cut
Proceedings of the forty-fifth annual ACM symposium on Theory of computing
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We show that any distribution on $\pmo^n$ that is $k$-wise independent fools any halfspace (a.k.a.~threshold) $h : \pmo^n \to \pmo$, i.e., any function of the form $h(x) = \sign(\sum_{i = 1}^n w_i x_i -\theta)$ where the $w_1,\ldots, w_n,\theta$ are arbitrary real numbers, with error $\eps$ for $k = O(\eps^{-2}\log^2(1/\eps))$. Our result is tight up to $\log(1/\eps)$ factors. Using standard constructions of $k$-wise independent distributions, we obtain the first explicit pseudorandom generators $G : \pmo^s \to \pmo^n$ that foolhalfspaces. Specifically, we fool halfspaces with error $\eps$ and seed length $s = k \cdot \log n = O(\log n \cdot \eps^{-2} \log^2(1/\eps))$.Our approach combines classical tools from real approximation theory with structural results on halfspaces by Servedio (Comput.~Complexity2007).