Agreement theorems for high dimensional expanders in the small soundness regime: the role of covers
Yotam Dikstein, Irit Dinur
TL;DR
This work analyzes agreement testing for high-dimensional expanders (HDXs) in the small-soundness regime and reveals that the feasibility of a strong 1%-agreement theorem is governed by topological covers of the underlying complex. The authors show a dichotomy: if the complex $X$ has no connected covers and satisfies swap-cosystolic expansion, then a 1% agreement implies a poly(1/ε)-sized list of global reconstructions that explain the local data (LD). If $X$ admits a connected cover, LD fails, but one can salvage a lift-decoding (LFD) statement by passing to a suitably chosen cover, yielding a global function on the cover that explains a constant fraction of the local data. The main theorem combines a multi-stage local-to-global construction: from 1% agreement to a structured family of per-face lists, passing to a face complex and then to a cover $Y$ of $X$, and finally aggregating to a global function on $Y(0)$; under swap-cosystolic expansion, this yields a polynomial-size cover and a global decoding function. The results apply to spherical buildings and Lubotzky–Sarnak–Venkatesh-type HDXs and have implications for derandomized direct-product encodings and PCP-related questions, highlighting how topological covers can be the right framework for understanding small-soundness phenomena in HDXs.
Abstract
Given a family $X$ of subsets of $[n]$ and an ensemble of local functions $\{f_s:s\toΣ\; | \; s\in X\}$, an agreement test is a randomized property tester that is supposed to test whether there is some global function $G:[n]\toΣ$ such that $f_s=G|_s$ for many sets $s$. A "classical" small-soundness agreement theorem is a list-decoding $(LD)$ statement, saying that \[\tag{$LD$} Agree(\{f_s\}) > \varepsilon \quad \Longrightarrow \quad \exists G^1,\dots, G^\ell,\quad P_s[f_s\overset{0.99}{\approx}G^i|_s]\geq poly(\varepsilon),\;i=1,\dots,\ell. \] Such a statement is motivated by PCP questions and has been shown in the case where $X=\binom{[n]}k$, or where $X$ is a collection of low dimensional subspaces of a vector space. In this work we study small the case of on high dimensional expanders $X$. It has been an open challenge to analyze their small soundness behavior. Surprisingly, the small soundness behavior turns out to be governed by the topological covers of $X$.We show that: 1. If $X$ has no connected covers, then $(LD)$ holds, provided that $X$ satisfies an additional expansion property. 2. If $X$ has a connected cover, then $(LD)$ necessarily fails. 3. If $X$ has a connected cover (and assuming the additional expansion property), we replace the $(LD)$ by a weaker statement we call lift-decoding: \[ \tag{$LFD$} Agree(\{f_s\})> \varepsilon \Longrightarrow \quad \exists\text{ cover }ρ:Y\twoheadrightarrow X,\text{ and }G:Y(0)\toΣ,\text{ such that }\] \[P_{\tilde s\twoheadrightarrow s}[f_s \overset{0.99}{\approx} G|_{\tilde s}] \geq poly(\varepsilon),\] where ${\tilde s\twoheadrightarrow s}$ means that $ρ(\tilde s)=s$. The additional expansion property is cosystolic expansion of a complex derived from $X$ holds for the spherical building and for quotients of the Bruhat-Tits building.