Estimating the Distance from Testable Affine-Invariant Properties

Let P be an affine invariant property of multivariate functions over a constant size finite field. We show that if P is locally testable with a constant number of queries, then one can estimate the distance of a function f from P with a constant number of queries. This was previously unknown even fo...

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Bibliographic Details
Published inAnnual Symposium on Foundations of Computer Science pp. 237 - 242
Main Authors Hatami, Hamed, Lovett, Shachar
Format Conference Proceeding
LanguageEnglish
Published IEEE 01.10.2013
Subjects
affine invariant properties
Computer science
Correlation
Educational institutions
Frequency modulation
higher-order fourier analysis
Polynomials
Probabilistic logic
property testing
Testing
Online AccessGet full text
ISSN0272-5428
DOI10.1109/FOCS.2013.33

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Summary:Let P be an affine invariant property of multivariate functions over a constant size finite field. We show that if P is locally testable with a constant number of queries, then one can estimate the distance of a function f from P with a constant number of queries. This was previously unknown even for simple properties such as cubic polynomials over the binary field. Our test is simple: take a restriction of f to a constant dimensional affine subspace, and measure its distance from P. We show that by choosing the dimension large enough, this approximates with high probability the global distance of f from P. 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].
ISSN:0272-5428
DOI:10.1109/FOCS.2013.33