Inequalities and tail bounds for elementary symmetric polynomial with
applications
release_emtktapwfnhodicgh3u4zmuh3e
by
Parikshit Gopalan, Amir Yehudayoff
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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