从可测仿射不变性质估计距离
Estimating the distance from testable affine-invariant properties
摘要：
设$\cal{P}$是函数$\mathbb{F}_p^n \to [R]$对于固定$p$和$R$的仿射不变性质。我们表明，如果$\cal{P}$是本地可测试的一个常数数量的查询，然后可以估计的距离从$\cal{P}$的查询的一个函数$f$。这是以前未知的，即使是简单的性质，如三次多项式在$\mathbb{F}_2$。我们的测试很简单：把$f$限制到一个常维仿射子空间，并测量它到$\cal{P}$的距离。我们表明，通过选择足够大的尺寸，这近似与高概率的全球距离的$f$从$\cP$。该分析结合了Fischer和Newman [SIAM J. Comp 2007]的方法，他们为图形属性建立了类似的结果，并结合了最近开发的高阶傅立叶分析工具，特别是Bhattacharyya等人[STOC 2013]开发的工具。
英文摘要：
Let $\cal{P}$ be an affine invariant property of functions $\mathbb{F}_p^n \to [R]$ for fixed $p$ and $R$. We show that if $\cal{P}$ is locally testable with a constant number of queries, then one can estimate the distance of a function $f$ from $\cal{P}$ with a constant number of queries. This was previously unknown even for simple properties such as cubic polynomials over $\mathbb{F}_2$. Our test is simple: take a restriction of $f$ to a constant dimensional affine subspace, and measure its distance from $\cal{P}$. We show that by choosing the dimension large enough, this approximates with high probability the global distance of $f$ from $\cP$. The analysis combines the approach of Fischer and Newman [SIAM J. Comp 2007] who established a similar result for graph properties, with recently developed tools in higher order Fourier analysis, in particular those developed in Bhattacharyya et al. [STOC 2013].
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