基于全局超压缩性的最优检验结果的改进
Improved Optimal Testing Results from Global Hypercontractivity
摘要：
测试低次多项式的问题由于其在理论计算机科学，特别是在复杂性理论中的重要性，多年来受到了极大的关注。该问题由三个参数指定：字段大小$q$、度$d$和邻近度参数$δ$，并且目标是设计对给定函数进行尽可能少的查询的测试器，该测试器能够区分给定函数具有至多$d$度的情况和给定函数是$δ$ -远离任何度$d$函数的情况。如果进行$O(q^d+1/δ)$查询（已知是必要的），则测试仪被称为最佳。对于尺寸为$q$的区域，Bhattacharyya等人首先证明了天然$t$ -平板测试仪对于$q=2$是最佳的，随后Haramaty等人证明了天然$t$-平板测试仪对于所有素数幂$q$是最佳的。然而，对字段大小的依赖是一个塔式函数。我们改进了上面的结果，表明字段大小的依赖性是多项式的。我们的方法也适用于更一般的设置提升仿射不变码，是基于研究的结构的错误的子空间的集合。即，子空间$A$使得$f|_{A}$的度数大于$d$。为此，我们观察到这些集合在格拉斯曼图的仿射版本中扩展得很差，并使用它通过全局超收缩性来建立它们的结构结果。然后，我们使用该结构对$f$执行局部校正。
英文摘要：
The problem of testing low-degree polynomials has received significant attention over the years due to its importance in theoretical computer science, and in particular in complexity theory. The problem is specified by three parameters: field size $q$, degree $d$ and proximity parameter $δ$, and the goal is to design a tester making as few as possible queries to a given function, which is able to distinguish between the case the given function has degree at most $d$, and the case the given function is $δ$-far from any degree $d$ function. A tester is called optimal if it makes $O(q^d+1/δ)$ queries (which are known to be necessary). For the field of size $q$, the natural $t$-flat tester was shown to be optimal first by Bhattacharyya et al. for $q=2$, and later by Haramaty et al. for all prime powers $q$. The dependency on the field size, however, is a tower-type function. We improve the results above, showing that the dependency on the field size is polynomial. Our approach also applies in the more general setting of lifted affine invariant codes, and is based on studying the structure of the collection of erroneous subspaces. i.e. subspaces $A$ such that $f|_{A}$ has degree greater than $d$. Towards this end, we observe that these sets are poorly expanding in the affine version of the Grassmann graph and use that to establish structural results on them via global hypercontractivity. We then use this structure to perform local correction on $f$.
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