Direct product testing

with Irit Dinur. CCC 2014.



A direct product is a function of the form g(x1,,xk)=(g1(x1),,gk(xk))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 22 queries, that is, a canonical two-query test distinguishes between direct products and functions that are 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 natural two query test for a given function f:[N]k[M]kf:[N]^k\to[M]^k

Two query direct product test: Choose x,yx,y that agree on a random set AA of tt coordinates and accept if f(x)A=f(y)Af(x)_A=f(y)_A.

We provide a comprehensive analysis of this test for all parameters N,M,k,tO(k)N,M,k,t\le O(k) and success probability δ>0\delta>0.

Our main result is that if a given function f:[N]k[M]kf:[N]^k\to[M]^k passes the test with probability δ1ε\delta \ge 1-\varepsilon then there is a direct product function gg such that Pr[f(x)=g(x)]1O(ε)\Pr[f(x)=g(x)]\ge 1-O(\varepsilon). This is the first result relating success in the (or any) test to the fraction of the domain on which ff is equal to a direct product function. This test has been analyzed in previous works for the case tkNt \ll k \ll N, and results show closeness of ff to a direct product under a less natural measure of “approximate agreement”.

In the small soundness “list-decoding” regime, we show that if the test above passes with probability δexp(k)\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 δexp(k3)\delta \ge \exp(-\sqrt[3]k) to the entire meaningful range of δ>exp(k)\delta>\exp(-k).