Characterizing Direct Product Testing via Coboundary Expansion

A d-dimensional simplicial complex X is said to support a direct product tester if any locally consistent function defined on its k-faces (where k ≪ d) necessarily come from a function over its vertices. More precisely, a direct product tester has a distribution µ over pairs of k-faces (A, A ′ ), and given query access to F : X(k) → 0, 1 k it samples (A, A ′ ) ∼ µ and checks that F [A]| A∩A ′ = F [A ′ ]| A∩A ′ . The tester should have (1) the "completeness property", meaning that any assignment F which is a direct product assignment passes the test with probability 1, and (2) the "soundness property", meaning that if F passes the test with probability s, then F must be correlated with a direct product function.

Dinur and Kaufman showed that a sufficiently good spectral expanding complex X admits a direct product tester in the "high soundness" regime where s is close to 1. They asked whether there are high dimensional expanders that support direct product tests in the "low soundness", when s is close to 0.

We give a characterization of high-dimensional expanders that support a direct product tester in the low soundness regime. We show that spectral expansion is insufficient, and the complex must additionally satisfy a variant of coboundary expansion, which we refer to as Unique-Games coboundary expanders. Conversely, we show that this property is also sufficient to get direct product testers. This property can be seen as a high-dimensional generalization of the standard notion of coboundary expansion over non-Abelian groups for 2-dimensional complexes. It asserts that any locally consistent Unique-Games instance obtained using the low-level faces of the complex, must admit a good global solution.