Testing Halfspaces

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 ǫ-far from any halfspace using only poly(1 ǫ ) 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 ...

• Type: Journal
• Year: 2007
• Where: ECCC
• Authors: Kevin Matulef, Ryan O'Donnell, Ronitt Rubinfeld, Rocco A. Servedio
• Keywords: Boolean Cube, ECCC 2007, Gaussian Distribution, Halfspace