On the Fourier spectrum of monotone functions

Abstract

In this paper, monotone Boolean functions are studied using harmonic analysis on the cube. The main result is that any monotone Boolean function has most of its power spectrum on its Fourier coefficients of “degree” at most O(=n) under any product distribution. This is similar to a result of Linial et al. [1993], which showed that AC functions have almost all of their power spectrum on the coefficients of degree, at most (log n), under the uniform distribution. As a consequence of the main result, the following two corollaries are obtained: —For any e . 0, monotone Boolean functions are PAC learnable with error e under product distributions in time 2O((1/e)n). —Any monotone Boolean function can be approximated within error e under product distributions by a non-monotone Boolean circuit of size 2O(1/en) and depth O(1/e =n). The learning algorithm runs in time subexponential as long as the required error is V(1/(=n log n)). It is shown that this is tight in the sense that for any subexponential time algorithm there is a monotone Boolean function for which this algorithm cannot approximate with error better than O(1/=n). The main result is also applied to other problems in learning and complexity theory. In learning theory, several polynomial-time algorithms for learning some classes of monotone Boolean functions, such as Boolean functions with O(logn/log log n) relevant variables, are presented. In complexity theory, some questions regarding monotone NP-complete problems are addressed.