On sums of locally testable affine invariant properties

Eli Ben-Sasson, Elena Grigorescu, Ghid Maatouk, Amir Shpilka, Madhu Sudan

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

Abstract

Affine-invariant properties are an abstract class of properties that generalize some central algebraic ones, such as linearity and low-degree-ness, that have been studied extensively in the context of property testing. Affine invariant properties consider functions mapping a big field double-struck F qn to the subfield double-struck Fq and include all properties that form an double-struck Fq-vector space and are invariant under affine transformations of the domain. Almost all the known locally testable affine-invariant properties have so-called "single-orbit characterizations" - namely they are specified by a single local constraint on the property, and the "orbit" of this constraint, i.e., translations of this constraint induced by affine-invariance. Single-orbit characterizations by a local constraint are also known to imply local testability. In this work we show that properties with single-orbit characterizations are closed under "summation". To complement this result, we also show that the property of being an n-variate low-degree polynomial over double-struck Fq has a single-orbit characterization (even when the domain is viewed as double-struck Fqn and so has very few affine transformations). As a consequence we find that the sum of any sparse affine-invariant property (properties satisfied by qO(n)-functions) with the set of degree d multivariate polynomials over double-struck F q has a single-orbit characterization (and is hence locally testable) when q is prime. We conclude with some intriguing questions/conjectures attempting to classify all locally testable affine-invariant properties.

Original languageEnglish
Title of host publicationApproximation, Randomization, and Combinatorial Optimization
Subtitle of host publicationAlgorithms and Techniques - 14th International Workshop, APPROX 2011 and 15th International Workshop, RANDOM 2011, Proceedings
Pages400-411
Number of pages12
DOIs
StatePublished - 2011
Externally publishedYes
Event14th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2011 and the 15th International Workshop on Randomization and Computation, RANDOM 2011 - Princeton, NJ, United States
Duration: 17 Aug 201119 Aug 2011

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume6845 LNCS
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349

Conference

Conference14th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2011 and the 15th International Workshop on Randomization and Computation, RANDOM 2011
Country/TerritoryUnited States
CityPrinceton, NJ
Period17/08/1119/08/11

Keywords

  • Direct sums
  • Error-correcting codes
  • Property testing
  • Symmetries

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