Testing Halfspaces
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070789_final.pdf
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Author(s)
Matulef, Kevin M.
O'Donnell, Ryan
Rubinfeld, Ronitt
Servedio, Rocco A.
Date Issued
February 2010
Journal
SIAM Journal on Computing
Publisher
Society for Industrial and Applied Mathematics
Citation
Matulef, Kevin et al. “Testing Halfspaces.” SIAM Journal on Computing 39.5 (2010): 2004.
Version
Author's final manuscript
Abstract
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 [dot] x-theta). We consider halfspaces over the continuous domain R[superscript n] (endowed with the standard multivariate Gaussian distribution) as well as halfspaces over the Boolean cube {-1,1}[superscript n] (endowed with the uniform distribution). In both cases we give an algorithm that distinguishes halfspaces from functions that are epsilon-far from any halfspace using only poly([fraction 1 over epsilon]) 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}[superscript 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}[superscript 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 [E. Fischer, G. Kindler, D. Ron, S. Safra, and A. Samorodnitsky, Proceedings of the 43rd IEEE Symposium on Foundations of Computer Science, 2002, pp. 103–112].
MIT Department
Massachusetts Institute of Technology. Computer Science and Artificial Intelligence Laboratory
Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science
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Creative Commons Attribution-Noncommercial-Share Alike 3.0
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DOI of Published Version
https://doi.org/10.1137/070707890
