TY - GEN

T1 - Testing halfspaces

AU - Matulef, Kevin

AU - O'Donnell, Ryan

AU - Rubinfeld, Ronitt

AU - Servedio, Rocco A.

PY - 2009

Y1 - 2009

N2 - 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].

AB - 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].

UR - http://www.scopus.com/inward/record.url?scp=70349147383&partnerID=8YFLogxK

U2 - 10.1137/1.9781611973068.29

DO - 10.1137/1.9781611973068.29

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AN - SCOPUS:70349147383

SN - 9780898716801

T3 - Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms

SP - 256

EP - 264

BT - Proceedings of the 20th Annual ACM-SIAM Symposium on Discrete Algorithms

PB - Association for Computing Machinery

T2 - 20th Annual ACM-SIAM Symposium on Discrete Algorithms

Y2 - 4 January 2009 through 6 January 2009

ER -