Paper Info
Title
Direct Product Testing
Abstract
A direct product function is a function of the form g(x_1, ldots, x_k)=(g_1(x_1), ldots, g_k(x_k)). We show that the direct product property is locally testable with two queries, that is, a canonical two-query test distinguishes between direct product functions and functions that are far from direct products with constant probability. This local testing question comes up naturally in the context of PCPs, where direct products play a prominent role for gap amplification. We consider the following natural two query test for a given function f:[N]^kto[M]^k begin{quote} {bf Two query direct product test}: Choose x, y that agree on a random set A of t coordinates and accept if f(x)_A=f(y)_A. End{quote} We provide a comprehensive analysis of this test for all parameters N, M, k, tle O(k) and success probability delta0. Our main result is that if a given function f:[N]^kto[M]^k passes the test with probability delta ge 1-eps then there is a direct product function g such that Pr[f(x)=g(x)]ge 1-O(eps). This is the first result relating success in the above (or any) test to the fraction of the domain on which f is equal to a direct product function. This test has been analyzed in previous works for the case t ll k ll N, and results show closeness of f to a direct product under a less natural measure of \"approximate agreement\". In the small soundness regime, we prove that if the test above passes with probability delta ge exp(-k), then the function agrees with a direct product function on local parts of the domain. This extends the previous range of parameters of delta ge exp(-sqrt[3]k) to the entire meaningful range of deltaexp(-k).
Year
DOI
Venue
2013
10.1109/CCC.2014.27
IEEE Conference on Computational Complexity
Keywords
DocType
Volume
parallel repetition, property testing, probabilistically checkable proofs,games,property testing,polynomials,computational complexity,decoding,game theory,testing,probability
Journal
20
ISSN
Citations 
PageRank 
1093-0159
9
0.59
References 
Authors
11
2
Name
Order
Citations
PageRank
Irit Dinur1118785.67
David Steurer293444.91