Evaluating spectral norms for constant depth circuits with symmetric gates

Abstract

The Fourier spectrum and its norms are given as explicit arithmetic expressions and evaluated, for Boolean functions computed by classes of constant depth, read-once circuits consisting of an arbitrary set of symmetric gates. Previous results of this nature estimate the spectralL 1 norm of functions computed by certain types of decision trees [20], [7], and in some cases, give randomizedprocedures that evaluate the spectrum by clever rounding [20]. One corollary of our results provides a large class ofAC 0 functions whose spectralL 1 norm is exponential, thus generalizing the single example of such a function given in [9]. This shows that almost every read-onceAC 0 function does not belong in the classPL 1 of functions with polynomially bounded spectral norms. Implications of our results and technique are discussed, for estimating the spectral norms ofany function in a constant depth circuit class, using the coding theoretic concept of weight distributions. Evaluating the spectral norms for any such function reduces to estimating certain non-trivial weight distributions of simple, linear codes.