Boolean Functions with Biased Inputs: Approximation and Noise Sensitivity

Abstract

This paper considers the problem of approximating a Boolean function f using another Boolean function from a specified class. Two classes of approximating functions are considered: k-juntas, and linear Boolean functions. The n input bits of the function are assumed to be independently drawn from a distribution that may be biased. The quality of approximation is measured by the mismatch probability between f and the approximating function g. For each class, the optimal approximation and the associated mismatch probability is characterized in terms of the biased Fourier expansion of f. The technique used to analyze the mismatch probability also yields an expression for the noise sensitivity of f in terms of the biased Fourier coefficients, under a general i.i.d. input perturbation model.