Toward a General Direct Product Testing Theorem

The direct product encoding of a string a∈ { 0,1}n on an underlying domain V⊆ (k[n]) is a function DPV(a) that gets as input a set S∈ V and outputs a restricted to S. In the direct product testing problem, we are given a function F:V→ { 0,1}k, and our goal is to test whether F is close to a direct product encoding—that is, whether there exists some a∈ { 0,1}n such that on most sets S, we have F(S)=DPV(a)(S). A natural test is as follows: select a pair (S,S′)∈ V according to some underlying distribution over V× V, query F on this pair, and check for consistency on their intersection. Note that the preceding distribution may be viewed as a weighted graph over the vertex set V and is referred to as a test graph. The testability of direct products was studied over various domains and test graphs: Dinur and Steurer (CCC’14) analyzed it when V equals the k-th slice of the Boolean hypercube and the test graph is a member of the Johnson graph family. Dinur and Kaufman (FOCS’17) analyzed it for the case where V is the set of faces of a Ramanujan complex, where in this case ∣V∣=Ok(n). In this article, we study the testability of direct products in a general setting, addressing the question: what properties of the domain and the test graph allow one to prove a direct product testing theorem? Towards this goal, we introduce the notion of coordinate expansion of a test graph. Roughly speaking, a test graph is a coordinate expander if it has global and local expansion, and has certain nice intersection properties on sampling. We show that whenever the test graph has coordinate expansion, it admits a direct product testing theorem. Additionally, for every k and n, we provide a direct product domain V⊆ (kn) of size n, called the sliding window domain, for which we prove direct product testability.