Towards a general direct product testing theorem

The Direct Product encoding of a string a ∈ {0, 1}ⁿ on an underlying domain V ⊆ ^([n_k^]), is a function DP_V (a) which 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, i.e., whether there exists some a ∈ {0, 1}ⁿ such that on most sets S, we have F(S) = DP_V (a)(S). A natural test is as follows: select a pair (S, S⁰) ∈ V according to some underlying distribution over V × V , query F on this pair, and check for consistency on their intersection. Note that the above 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 = O_k(n). In this paper, 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 then it admits a direct product testing theorem. Additionally, for every k and n we provide a direct product domain V ⊆ ^{(n _k)} of size n, called the Sliding Window domain for which we prove direct product testability.