Threshold circuits of bounded depth
Journal of Computer and System Sciences
Active Learning Using Arbitrary Binary Valued Queries
Machine Learning
Property testing and its connection to learning and approximation
Journal of the ACM (JACM)
On Restricted-Focus-of-Attention Learnability of Boolean Functions
Machine Learning - Special issue on the ninth annual conference on computational theory (COLT '96)
Testing Basic Boolean Formulae
SIAM Journal on Discrete Mathematics
FOCS '02 Proceedings of the 43rd Symposium on Foundations of Computer Science
SIAM Journal on Discrete Mathematics
Every Linear Threshold Function has a Low-Weight Approximator
CCC '06 Proceedings of the 21st Annual IEEE Conference on Computational Complexity
Every Linear Threshold Function has a Low-Weight Approximator
Computational Complexity
Optimal Inapproximability Results for MAX-CUT and Other 2-Variable CSPs?
SIAM Journal on Computing
Testing for Concise Representations
FOCS '07 Proceedings of the 48th Annual IEEE Symposium on Foundations of Computer Science
STOC '08 Proceedings of the fortieth annual ACM symposium on Theory of computing
SFCS '90 Proceedings of the 31st Annual Symposium on Foundations of Computer Science
Property testing: a learning theory perspective
COLT'07 Proceedings of the 20th annual conference on Learning theory
Green's conjecture and testing linear-invariant properties
Proceedings of the forty-first annual ACM symposium on Theory of computing
Testing Fourier Dimensionality and Sparsity
ICALP '09 Proceedings of the 36th International Colloquium on Automata, Languages and Programming: Part I
APPROX '09 / RANDOM '09 Proceedings of the 12th International Workshop and 13th International Workshop on Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques
Testing Boolean function isomorphism
APPROX/RANDOM'10 Proceedings of the 13th international conference on Approximation, and 14 the International conference on Randomization, and combinatorial optimization: algorithms and techniques
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
Testing juntas: a brief survey
Property testing
Invariance in property testing
Property testing
Testing (subclasses of) halfspaces
Property testing
Testing juntas: a brief survey
Property testing
Invariance in property testing
Property testing
Testing (subclasses of) halfspaces
Property testing
Hi-index | 0.00 |
This paper addresses the problem of testing whether a Boolean-valued function f is a halfspace, i.e. a function of the form f(x) = sgn(w · x - θ). We consider halfspaces over the continuous domain Rn (endowed with the standard multivariate Gaussian distribution) as well as halfspaces over the Boolean cube {−1, 1}n (endowed with the uniform distribution). In both cases we give an algorithm that distinguishes halfspaces from functions that are ε-far from any halfspace using only poly(1/ε) queries, independent of the dimension n. Two simple structural results about halfspaces are at the heart of our approach for the Gaussian distribution: the first gives an exact relationship between the expected value of a halfspace f and the sum of the squares of f's degree-1 Hermite coefficients, and the second shows that any function that approximately satisfies this relationship is close to a halfspace. We prove analogous results for the Boolean cube {−1, 1}n (with Fourier coefficients in place of Hermite coefficients) for balanced halfspaces in which all degree-1 Fourier coefficients are small. Dealing with general halfspaces over {−1, 1}n poses significant additional complications and requires other ingredients. These include "cross-consistency" versions of the results mentioned above for pairs of halfspaces with the same weights but different thresholds; new structural results relating the largest degree-1 Fourier coefficient and the largest weight in unbalanced halfspaces; and algorithmic techniques from recent work on testing juntas [FKR+02].