Lower Bounds on Same-Set Inner Product in Correlated Spaces
Let P be a probability distribution over a finite alphabet Omega^L with all L marginals equal. Let X^(1), ..., X^(L), where X^(j) = (X_1^(j), ..., X_n^(j)) be random vectors such that for every coordinate i in [n] the tuples (X_i^(1), ..., X_i^(L)) are i.i.d. according to P.
The question we address is: does there exist a function c_P independent of n such that for every f: Omega^n -> [0, 1] with E[f(X^(1))] = m > 0 we have E[f(X^(1)) * ... * f(X^(n))] > c_P(m) > 0?
We settle the question for L=2 and when L>2 and P has bounded correlation smaller than 1.
same set hitting
product spaces
correlation
lower bounds
34:1-34:11
Regular Paper
Jan
Hazla
Jan Hazla
Thomas
Holenstein
Thomas Holenstein
Elchanan
Mossel
Elchanan Mossel
10.4230/LIPIcs.APPROX-RANDOM.2016.34
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode