On sums of locally testable affine invariant properties

Authors:
Eli Ben-Sasson;Elena Grigorescu;Ghid Maatouk;Amir Shpilka;Madhu Sudan
Affiliations:
Department of Computer Science, Technion -Israel Institute of Technology, Haifa, Israel;Georgia Tech, Atlanta, GA;School of Computer and Communications Sciences, EPFL, Switzerland;Faculty of Computer Science, Technion - Israel Institute of Technology, Haifa, Israel and Microsoft Research, Cambridge, MA;Microsoft Research New England, Cambridge, Massachusetts
Venue:
APPROX'11/RANDOM'11 Proceedings of the 14th international workshop and 15th international conference on Approximation, randomization, and combinatorial optimization: algorithms and techniques
Year:
2011

Citing 17
Cited 0

Probabilistic checking of proofs: a new characterization of NP

Journal of the ACM (JACM)
Proof verification and the hardness of approximation problems

Journal of the ACM (JACM)
Property testing and its connection to learning and approximation

Journal of the ACM (JACM)
Private information retrieval

Journal of the ACM (JACM)
On the efficiency of local decoding procedures for error-correcting codes

STOC '00 Proceedings of the thirty-second annual ACM symposium on Theory of computing
Robust Characterizations of Polynomials withApplications to Program Testing

SIAM Journal on Computing
A combinatorial characterization of the testable graph properties: it's all about regularity

Proceedings of the thirty-eighth annual ACM symposium on Theory of computing
Graph limits and parameter testing

Proceedings of the thirty-eighth annual ACM symposium on Theory of computing
Testing Polynomials over General Fields

SIAM Journal on Computing
Algebraic property testing: the role of invariance

STOC '08 Proceedings of the fortieth annual ACM symposium on Theory of computing
2-Transitivity Is Insufficient for Local Testability

CCC '08 Proceedings of the 2008 IEEE 23rd Annual Conference on Computational Complexity
Testing low-degree polynomials over prime fields

Random Structures & Algorithms
Succinct Representation of Codes with Applications to Testing

APPROX '09 / RANDOM '09 Proceedings of the 12th International Workshop and 13th International Workshop on Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques
On Locally Decodable Codes, Self-Correctable Codes, and t-Private PIR

Algorithmica
Rank bounds for design matrices with applications to combinatorial geometry and locally correctable codes

Proceedings of the forty-third annual ACM symposium on Theory of computing
Symmetric LDPC Codes are not Necessarily Locally Testable

CCC '11 Proceedings of the 2011 IEEE 26th Annual Conference on Computational Complexity
Testing Reed-Muller codes

IEEE Transactions on Information Theory

Quantified Score

Hi-index	0.00

Visualization

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 Fqn to the subfield Fq and include all properties that form an 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 affineinvariance. 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 Fq has a single-orbit characterization (even when the domain is viewed as 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 Fq 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.