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 min-wise 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 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 $x \in \mathbb{R}^n$ which we believe to be of independent interest. We show that if $|S_k(x)|,|S_{k+1}(x)|$ are small relative to $|S_{k-1}(x)|$ 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 e-prints
- Pub Date:
- February 2014
- DOI:
- 10.48550/arXiv.1402.3543
- arXiv:
- arXiv:1402.3543
- Bibcode:
- 2014arXiv1402.3543G
- Keywords:
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- Computer Science - Computational Complexity;
- 68Q87;
- 68W20