Low-degree Pseudo-Boolean Function Recovery Using Codes

Abstract

Pseudo-Boolean functions are functions whose input variables are binary and output is in the real numbers. These functions show up in many different applications in computer science, finance and economics to name a few. Pseudo-Boolean functions lend themselves to a spectral representation, which is closely related to the Walsh-Hadamard Transform from signal processing. In some problems, the coefficients of the spectral representation are active only on the low-degree terms. In this work, we present a method for computationally-efficient recovery of these low-degree coefficients. Our method is based on evaluating the input pseudo-Boolean function at points given by the codewords of a codebook, and then performing a Walsh-Hadamard Transform on the resulting signal. Codes having high rates and good minimum distance properties yield sets of evaluations points whose size is close to the number of low-degree coefficients. In particular perfect codes, such as Hamming Codes or the Golay Code, enable efficient recovery with optimal number of evaluations of the function.