A direct product 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 \(2\) 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 \(f:[N]^k\to[M]^k\)
Two query direct product test: Choose \(x,y\) that agree on a random set \(A\) of \(t\) coordinates and accept if \(f(x)_A=f(y)_A\).
We provide a comprehensive analysis of this test for all parameters \(N,M,k,t\le O(k)\) and success probability \(\delta>0\).
Our main result is that if a given function \(f:[N]^k\to[M]^k\) passes the test with probability \(\delta \ge 1-\varepsilon\) then there is a direct product function \(g\) such that \(\Pr[f(x)=g(x)]\ge 1-O(\varepsilon)\). This is the first result relating success in the (or any) test to the fraction of the domain on which \(f\) 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 “list-decoding” regime, we show 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 \(\delta \ge \exp(-\sqrtk)\) to the entire meaningful range of \(\delta>\exp(-k)\).