ECCC-Report TR13-087https://eccc.weizmann.ac.il/report/2013/087Comments and Revisions published for TR13-087en-usThu, 13 Jun 2013 08:59:36 +0300
Paper TR13-087
| Estimating the distance from testable affine-invariant properties |
Shachar Lovett,
Hamed Hatami
https://eccc.weizmann.ac.il/report/2013/087Let $\cal{P}$ 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].
Thu, 13 Jun 2013 08:59:36 +0300https://eccc.weizmann.ac.il/report/2013/087