分层集合系统上的一致性检验定理
Agreement testing theorems on layered set systems
摘要：
我们引入了一个分层子集的框架，并给出了一个充分条件时，一个集系统支持的协议测试。一致性测试是一种特定类型的属性测试，它概括了PCP测试，如平面与平面测试。以前的工作表明，高维扩展是有用的协议测试。我们将这些结果扩展到更一般的家庭的子集，超越单纯复形。这些措施包括-协议测试集系统的集是面对高维扩展。我们的新的测试适用于所有维度的复杂的情况下，双方的扩张和单边partite扩张。这改进和扩展了Dinur和Kaufman的早期工作（FOCS 2017），并适用于拟阵，以及潜在的许多其他复形。- 集合是高维展开式中顶点邻域的集合系统的一致性检验。这个族类似于在PCP定理的间隙放大证明中使用的扩展邻域族。这个集合系统是相当自然的，但不坐在一个单纯的复杂，并证明了我们的证明技术的一些通用性。- 子空间族（也称为格拉斯曼偏序集）的一致性检验。这扩展了经典的低次一致性测试，超出了低次多项式的设置。我们的分析依赖于一个新的随机行走单纯复合物，我们称之为“互补随机行走”，这可能是独立的利益。这种随机游动将图上的非惰性随机游动推广到更高的维度，并且比以前研究的单纯复形上的随机游动具有更好的扩展性。
英文摘要：
We introduce a framework of layered subsets, and give a sufficient condition for when a set system supports an agreement test. Agreement testing is a certain type of property testing that generalizes PCP tests such as the plane vs. plane test. Previous work has shown that high dimensional expansion is useful for agreement tests. We extend these results to more general families of subsets, beyond simplicial complexes. These include - Agreement tests for set systems whose sets are faces of high dimensional expanders. Our new tests apply to all dimensions of complexes both in case of two-sided expansion and in the case of one-sided partite expansion. This improves and extends an earlier work of Dinur and Kaufman (FOCS 2017) and applies to matroids, and potentially many additional complexes. - Agreement tests for set systems whose sets are neighborhoods of vertices in a high dimensional expander. This family resembles the expander neighborhood family used in the gap-amplification proof of the PCP theorem. This set system is quite natural yet does not sit in a simplicial complex, and demonstrates some versatility in our proof technique. - Agreement tests on families of subspaces (also known as the Grassmann poset). This extends the classical low degree agreement tests beyond the setting of low degree polynomials. Our analysis relies on a new random walk on simplicial complexes which we call the ``complement random walk'' and which may be of independent interest. This random walk generalizes the non-lazy random walk on a graph to higher dimensions, and has significantly better expansion than previously-studied random walks on simplicial complexes.
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