Testing monotonicity over graph products
Random Structures & Algorithms
Property Testing: A Learning Theory Perspective
Foundations and Trends® in Machine Learning
Tolerant Linearity Testing and Locally Testable Codes
APPROX '09 / RANDOM '09 Proceedings of the 12th International Workshop and 13th International Workshop on Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques
Algorithmic and Analysis Techniques in Property Testing
Foundations and Trends® in Theoretical Computer Science
Approximating the distance to monotonicity in high dimensions
ACM Transactions on Algorithms (TALG)
Distribution-free testing algorithms for monomials with a sublinear number of queries
APPROX/RANDOM'10 Proceedings of the 13th international conference on Approximation, and 14 the International conference on Randomization, and combinatorial optimization: algorithms and techniques
The Journal of Machine Learning Research
Testing the lipschitz property over product distributions with applications to data privacy
TCC'13 Proceedings of the 10th theory of cryptography conference on Theory of Cryptography
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We consider the problem of distribution-free property-testing of functions. In this setting of property-testing, the distance between functions is measured with respect to a fixed but unknown distribution $D$ on the domain. The testing algorithms are given oracle access to random sampling from the domain according to this distribution $D$. This notion of distribution-free testing was previously defined, but no distribution-free property-testing algorithm was known for any (non-trivial) property. We present the first such distribution-free property-testing algorithms for two of the central problems in this field. The testers are obtained by extending some known results (from “standard,” uniform distribution, property-testing): (1) A distribution-free testing algorithm for low-degree multivariate polynomials with query complexity $O(d^2 + d \cdot \epsilon^{-1})$, where $d$ is the total degree of the polynomial. The same approach that is taken for the distribution-free testing of low-degree polynomials is shown to apply also to several other problems; (2) a distribution-free monotonicity testing algorithm for functions $f:[n]^d \rightarrow A$ for low dimensions (e.g., when $d$ is a constant) with query complexity similar to the one achieved in the uniform setting. On the negative side, we prove an exponential gap between the query complexity required for uniform and distribution-free monotonicity testing in the high-dimensional case.