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Evaluating spectral norms for constant depth circuits with symmetric gates

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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.

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Sitharam, M. Evaluating spectral norms for constant depth circuits with symmetric gates. Comput Complexity 5, 167–189 (1995). https://doi.org/10.1007/BF01268144

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