Testing ±1-weight halfspace

Kevin Matulef; Ryan O'Donnell; Ronitt Rubinfeld; Rocco A. Servedio

doi:10.1007/978-3-642-03685-9_48

Testing ±1-weight halfspace

Kevin Matulef^*, Ryan O'Donnell, Ronitt Rubinfeld, Rocco A. Servedio

^*Corresponding author for this work

School of Computer Science

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

Abstract

We consider the problem of testing whether a Boolean function f : { - 1, 1} ⁿ →{- 1, 1} is a ±1-weight halfspace, i.e. a function of the form f(x) = sgn(w ₁ x ₁ + w ₂ x ₂ + ⋯ + w _n x _n) where the weights w _i take values in { - 1, 1}. We show that the complexity of this problem is markedly different from the problem of testing whether f is a general halfspace with arbitrary weights. While the latter can be done with a number of queries that is independent of n [7], to distinguish whether f is a ±-weight halfspace versus ∈-far from all such halfspaces we prove that nonadaptive algorithms must make Ω(log n) queries. We complement this lower bound with a sublinear upper bound showing that O(poly queries suffice.

Original language	English
Title of host publication	Approximation, Randomization, and Combinatorial Optimization
Subtitle of host publication	Algorithms and Techniques - 12th International Workshop, APPROX 2009 and 13th International Workshop, RANDOM 2009, Proceedings
Pages	646-657
Number of pages	12
DOIs	https://doi.org/10.1007/978-3-642-03685-9_48
State	Published - 2009
Event	12th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2009 and 13th International Workshop on Randomization and Computation, RANDOM 2009 - Berkeley, CA, United States Duration: 21 Aug 2009 → 23 Aug 2009

Publication series

Name	Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume	5687 LNCS
ISSN (Print)	0302-9743
ISSN (Electronic)	1611-3349

Conference

Conference	12th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2009 and 13th International Workshop on Randomization and Computation, RANDOM 2009
Country/Territory	United States
City	Berkeley, CA
Period	21/08/09 → 23/08/09

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10.1007/978-3-642-03685-9_48

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Matulef, K., O'Donnell, R., Rubinfeld, R., & Servedio, R. A. (2009). Testing ±1-weight halfspace. In Approximation, Randomization, and Combinatorial Optimization: Algorithms and Techniques - 12th International Workshop, APPROX 2009 and 13th International Workshop, RANDOM 2009, Proceedings (pp. 646-657). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 5687 LNCS). https://doi.org/10.1007/978-3-642-03685-9_48

Matulef, Kevin ; O'Donnell, Ryan ; Rubinfeld, Ronitt et al. / Testing ±1-weight halfspace. Approximation, Randomization, and Combinatorial Optimization: Algorithms and Techniques - 12th International Workshop, APPROX 2009 and 13th International Workshop, RANDOM 2009, Proceedings. 2009. pp. 646-657 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)).

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title = "Testing ±1-weight halfspace",

abstract = "We consider the problem of testing whether a Boolean function f : { - 1, 1} n →{- 1, 1} is a ±1-weight halfspace, i.e. a function of the form f(x) = sgn(w 1 x 1 + w 2 x 2 + ⋯ + w n x n) where the weights w i take values in { - 1, 1}. We show that the complexity of this problem is markedly different from the problem of testing whether f is a general halfspace with arbitrary weights. While the latter can be done with a number of queries that is independent of n [7], to distinguish whether f is a ±-weight halfspace versus ∈-far from all such halfspaces we prove that nonadaptive algorithms must make Ω(log n) queries. We complement this lower bound with a sublinear upper bound showing that O(poly queries suffice.",

author = "Kevin Matulef and Ryan O'Donnell and Ronitt Rubinfeld and Servedio, {Rocco A.}",

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Matulef, K, O'Donnell, R, Rubinfeld, R & Servedio, RA 2009, Testing ±1-weight halfspace. in Approximation, Randomization, and Combinatorial Optimization: Algorithms and Techniques - 12th International Workshop, APPROX 2009 and 13th International Workshop, RANDOM 2009, Proceedings. Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 5687 LNCS, pp. 646-657, 12th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2009 and 13th International Workshop on Randomization and Computation, RANDOM 2009, Berkeley, CA, United States, 21/08/09. https://doi.org/10.1007/978-3-642-03685-9_48

Testing ±1-weight halfspace. / Matulef, Kevin; O'Donnell, Ryan; Rubinfeld, Ronitt et al.
Approximation, Randomization, and Combinatorial Optimization: Algorithms and Techniques - 12th International Workshop, APPROX 2009 and 13th International Workshop, RANDOM 2009, Proceedings. 2009. p. 646-657 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 5687 LNCS).

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

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Matulef K, O'Donnell R, Rubinfeld R, Servedio RA. Testing ±1-weight halfspace. In Approximation, Randomization, and Combinatorial Optimization: Algorithms and Techniques - 12th International Workshop, APPROX 2009 and 13th International Workshop, RANDOM 2009, Proceedings. 2009. p. 646-657. (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)). doi: 10.1007/978-3-642-03685-9_48

Testing ±1-weight halfspace

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