Low Acceptance Agreement Tests via Bounded-Degree Symplectic HDXs
Yotam Dikstein, Irit Dinur, Alexander Lubotzky
TL;DR
The paper constructs bounded-degree, linear-size high dimensional expanders arising as quotients of the affine symplectic building \tilde{C}_g(\mathbb{Q}_p) and proves these complexes have no small connected covers. Leveraging swap cocycle expansion and a local-to-global coboundary framework, the authors derive a 1% agreement test for derandomized direct product testing, improving previous bounds that required SL-type (type A) structures. A key innovation is replacing SL-based quotients with symplectic-type quotients, made possible by the congruence subgroup property for Sp(2g,\mathbb{Q}_p), and by extending swap expansion techniques to the symplectic setting. The practical impact lies in providing a polynomial-time, explicit construction of bounded-degree HDXs that enable strong low-acceptance agreement testing, with potential implications for PCPs and derandomized testing within complexity theory.
Abstract
We solve the derandomized direct product testing question in the low acceptance regime, by constructing new high dimensional expanders that have no small connected covers. We show that our complexes have swap cocycle expansion, which allows us to deduce the agreement theorem by relying on previous work. Derandomized direct product testing, also known as agreement testing, is the following problem. Let X be a family of k-element subsets of [n] and let $\{f_s:s\toΣ\}_{s\in X}$ be an ensemble of local functions, each defined over a subset $s\subset [n]$. Suppose that we run the following so-called agreement test: choose a random pair of sets $s_1,s_2\in X$ that intersect on $\sqrt k$ elements, and accept if $f_{s_1},f_{s_2}$ agree on the elements in $s_1\cap s_2$. We denote the success probability of this test by $Agr(\{f_s\})$. Given that $Agr(\{f_s\})=ε>0$, is there a global function $G:[n]\toΣ$ such that $f_s = G|_s$ for a non-negligible fraction of $s\in X$ ? We construct a family X of k-subsets of $[n]$ such that $|X| = O(n)$ and such that it satisfies the low acceptance agreement theorem. Namely, $Agr (\{f_s\}) > ε\; \; \longrightarrow$ there is a function $G:[n]\toΣ$ such that $\Pr_s[f_s\overset{0.99}{\approx} G|_s]\geq poly(ε)$. A key idea is to replace the well-studied LSV complexes by symplectic high dimensional expanders (HDXs). The family X is just the k-faces of the new symplectic HDXs. The later serve our needs better since their fundamental group satisfies the congruence subgroup property, which implies that they lack small covers. We also give a polynomial-time algorithm to construct this family of symplectic HDXs.