On spectral estimators of Boolean functions

The problem of estimating the Fourier spectra of Boolean functions using noisy non-uniformly drawn random examples is considered. In particular, arbitrary product distributions on the n-dimensional attribute vectors are assumed. The attributes are disturbed by noise also following a product distribution. Under these conditions the problem of estimating the Fourier spectra is considered. A general expression is derived that allows the construction of estimators of the Fourier spectra. This results can be applied to learn functions that are concentrated on the lower part of their spectra. As an application of the presented results an algorithm is shown that infers the relevant variables of so-called 1-low Boolean juntas.