Characterizing Direct Product Testing via Coboundary Expansion
Mitali Bafna, Dor Minzer
TL;DR
The paper addresses the problem of testing direct product encodings on $d$-dimensional simplicial complexes, distinguishing between high and low soundness regimes. It introduces Unique-Games coboundary expansion as a necessary and sufficient condition for enabling direct product testers in the low-soundness regime, showing that spectral expansion alone is insufficient. The authors develop a reduction to list agreement testing and leverage non-Abelian coboundary concepts to achieve global structure from locally consistent assignments. They prove both necessity and sufficiency of UG coboundary expansion for low-soundness direct product testing, and demonstrate the applicability to known complexes such as Johnson and Grassmann while identifying limits for LSV complexes. This work advances our understanding of high-dimensional expanders as robust testable encodings, with implications for PCPs, hardness amplification, and property testing in higher dimensions.
Abstract
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 $k\ll d$) 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 $F\colon X(k)\to\{0,1\}^k$ it samples $(A,A')\sim μ$ and checks that $F[A]|_{A\cap A'} = F[A']|_{A\cap A'}$. 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 \emph{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.