Inequalities and tail bounds for elementary symmetric polynomial with applications
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 minwise independent hashing. For random variables whose absolute value is bounded by $1$, we give an error bound of the form $\sigma^{\Omega(k)}$ where $k$ is the amount of independence and $\sigma^2$ 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 minwise independent hash functions and pseudorandom generators for combinatorial rectangles due to Gopalan et al., which also slightly improves their seedlength. Our proof relies on a new analytic inequality for the elementary symmetric polynomials $S_k(x)$ for $x \in \mathbb{R}^n$ which we believe to be of independent interest. We show that if $S_k(x),S_{k+1}(x)$ are small relative to $S_{k1}(x)$ for some $k>0$ then $S_\ell(x)$ is also small for all $\ell > k$. From these, we derive tail bounds for the elementary symmetric polynomials when the inputs are only $k$wise independent.
 Publication:

arXiv eprints
 Pub Date:
 February 2014
 arXiv:
 arXiv:1402.3543
 Bibcode:
 2014arXiv1402.3543G
 Keywords:

 Computer Science  Computational Complexity;
 68Q87;
 68W20