## Abstract

A signed majority function is a linear threshold function f : {+1, -1} ^{n} → {+1, -1} of the form f(x) = sign(∑_{i=1} ^{n} σ_{i}x_{¡}) where each σ_{i} ∈ {+1, -1}. Signed majority functions are a highly symmetrical subclass of the class of all linear threshold functions, which are functions of the form sign(∑_{i=1}^{n} W_{i}X _{i} - θ) for arbitrary real w_{i}, θ. We study the query complexity of testing whether an unknown f : {+1, -1}^{n} → {+1, -1} is a signed majority function versus ε-far from every signed majority function. While it is known [26] that the broader class of all linear threshold functions is testable with poly(1/ε) queries (independent of n), prior to our work the best upper bound for signed majority functions was O(√n)·poly(1/ε) queries (via a non-adaptive algorithm), and the best lower bound was Ω(log n) queries for non-adaptive algorithms [27]. As our main results we exponentially improve both these prior bounds for testing signed majority functions: • (Upper bound) We give a poly(log n, 1/ε)-query adaptive algorithm (which is computationally efficient) for this testing problem; • (Lower bound) We show that any non-adaptive algorithm for testing the class of signed majorities to constant accuracy must make n ^{Ω(1)} queries. This directly implies a lower bound of Ω(log n) queries for any adaptive algorithm. Our testing algorithm performs a sequence of restrictions together with consistency checks to ensure that each successive restriction is "compatible" with the function prior to restriction. This approach is used to transform the original n-variable testing problem into a testing problem over poly(log n, 1/ε) variables where a simple direct method can be applied. Analysis of the degree-1 Fourier coefficients plays an important role in our proofs.

Original language | English |
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Title of host publication | Proceedings of the 24th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2013 |

Publisher | Association for Computing Machinery |

Pages | 1319-1336 |

Number of pages | 18 |

ISBN (Print) | 9781611972511 |

DOIs | |

State | Published - 2013 |

Event | 24th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2013 - New Orleans, LA, United States Duration: 6 Jan 2013 → 8 Jan 2013 |

### Publication series

Name | Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms |
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### Conference

Conference | 24th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2013 |
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Country/Territory | United States |

City | New Orleans, LA |

Period | 6/01/13 → 8/01/13 |