Constant depth circuits, Fourier transform, and learnability

Nathan Linial, Yishay Mansour, Noam Nisan
1993 Journal of the ACM  
In this paper, Boolean functions in ,4C0 are studied using harmonic analysis on the cube. The main result is that an ACO Boolean function has almost all of its "power spectrum" on the low-order coefficients. An important ingredient of the proof is Hastad's switching lemma [8]. This result implies several new properties of functions in -4C[': Functions in AC() have low "average sensitivity;" they may be approximated well by a real polynomial of low degree and they cannot be pseudorandom function
more » ... generators. Perhaps the most interesting application is an O(n POIYIOg(n ')-time algorithm for learning functions in ACO. The algorithm observes the behavior of an AC'" function on O(nPO'Y'Og(n)) randomly chosen inputs, and derives a good approximation for the Fourier transform of the function. This approximation allows the algorithm to predict, with high probability, the value of the function on other randomly chosen inputs. A preliminary version of this paper was published in
doi:10.1145/174130.174138 fatcat:iwwlqkfxazgijjobyhhevsk4su