A hard-core predicate for all one-way functions
STOC '89 Proceedings of the twenty-first annual ACM symposium on Theory of computing
Learning decision trees using the Fourier spectrum
STOC '91 Proceedings of the twenty-third annual ACM symposium on Theory of computing
Property testing and its connection to learning and approximation
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Robust Characterizations of Polynomials withApplications to Program Testing
SIAM Journal on Computing
The difficulty of testing for isomorphism against a graph that is given in advance
STOC '04 Proceedings of the thirty-sixth annual ACM symposium on Theory of computing
Journal of Computer and System Sciences - Special issue on FOCS 2002
SODA '06 Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithm
Testing versus Estimation of Graph Properties
SIAM Journal on Computing
Testing for Concise Representations
FOCS '07 Proceedings of the 48th Annual IEEE Symposium on Foundations of Computer Science
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
A Combinatorial Characterization of the Testable Graph Properties: It's All About Regularity
SIAM Journal on Computing
Algorithmic and Analysis Techniques in Property Testing
Foundations and Trends® in Theoretical Computer Science
A Unified Framework for Testing Linear-Invariant Properties
FOCS '10 Proceedings of the 2010 IEEE 51st Annual Symposium on Foundations of Computer Science
Invariance in property testing
Property testing
Every property of hyperfinite graphs is testable
Proceedings of the forty-third annual ACM symposium on Theory of computing
Testing Fourier Dimensionality and Sparsity
SIAM Journal on Computing
Junto-Symmetric Functions, Hypergraph Isomorphism and Crunching
CCC '12 Proceedings of the 2012 IEEE Conference on Computational Complexity (CCC)
Partially Symmetric Functions Are Efficiently Isomorphism-Testable
FOCS '12 Proceedings of the 2012 IEEE 53rd Annual Symposium on Foundations of Computer Science
Every locally characterized affine-invariant property is testable
Proceedings of the forty-fifth annual ACM symposium on Theory of computing
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A function $f:\mathbb{F}_2^n \to \{-1,1\}$ is called linear-isomorphic to g if f=g∘A for some non-singular matrix A. In the g-isomorphism problem, we want a randomized algorithm that distinguishes whether an input function f is linear-isomorphic to g or far from being so. We show that the query complexity to test g-isomorphism is essentially determined by the spectral norm of g. That is, if g is close to having spectral norm s, then we can test g-isomorphism with poly(s) queries, and if g is far from having spectral norm s, then we cannot test g-isomorphism with o(logs) queries. The upper bound is almost tight since there is indeed a function g close to having spectral norm s whereas testing g-isomorphism requires Ω(s) queries. As far as we know, our result is the first characterization of this type for functions. Our upper bound is essentially the Kushilevitz-Mansour learning algorithm, modified for use in the implicit setting. Exploiting our upper bound, we show that any property is testable if it can be well-approximated by functions with small spectral norm. We also extend our algorithm to the setting where A is allowed to be singular.