@inproceedings{7e41b4ae7e51442eb937dd8a1fa39de2,
title = "On sums of locally testable affine invariant properties",
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.",
keywords = "Direct sums, Error-correcting codes, Property testing, Symmetries",
author = "Eli Ben-Sasson and Elena Grigorescu and Ghid Maatouk and Amir Shpilka and Madhu Sudan",
year = "2011",
doi = "10.1007/978-3-642-22935-0_34",
language = "אנגלית",
isbn = "9783642229343",
series = "Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)",
pages = "400--411",
booktitle = "Approximation, Randomization, and Combinatorial Optimization",
note = "null ; Conference date: 17-08-2011 Through 19-08-2011",
}