Inequalities and tail bounds for elementary symmetric polynomial with applications release_emtktapwfnhodicgh3u4zmuh3e

by Parikshit Gopalan, Amir Yehudayoff

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2014  

Abstract

We study the extent of independence needed to approximate the product of bounded random variables in expectation, a natural question that has applications in pseudorandomness and min-wise independent hashing. For random variables whose absolute value is bounded by 1, we give an error bound of the form σ^Ω(k) where k is the amount of independence and σ^2 is the total variance of the sum. Previously known bounds only applied in more restricted settings, and were quanitively weaker. We use this to give a simpler and more modular analysis of a construction of min-wise independent hash functions and pseudorandom generators for combinatorial rectangles due to Gopalan et al., which also slightly improves their seed-length. Our proof relies on a new analytic inequality for the elementary symmetric polynomials S_k(x) for x ∈R^n which we believe to be of independent interest. We show that if |S_k(x)|,|S_k+1(x)| are small relative to |S_k-1(x)| for some k>0 then |S_ℓ(x)| is also small for all ℓ > k. From these, we derive tail bounds for the elementary symmetric polynomials when the inputs are only k-wise independent.
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Date   2014-02-14
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arXiv  1402.3543v1
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