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Polynomial threshold functions, AC functions and spectrum norms

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2 Author(s)
Bruck, J. ; IBM Almaden Res. Center, San Jose, CA, USA ; Smolensky, Roman

The class of polynomial-threshold functions is studied using harmonic analysis, and the results are used to derive lower bounds related to AC0 functions. A Boolean function is polynomial threshold if it can be represented as a sign function of a sparse polynomial (one that consists of a polynomial number of terms). The main result is that polynomial-threshold functions can be characterized by means of their spectral representation. In particular, it is proved that a Boolean function whose L1 spectral norm is bounded by a polynomial in n is a polynomial-threshold function, and that a Boolean function whose L-1 spectral norm is not bounded by a polynomial in n is not a polynomial-threshold function. Some results for AC0 functions are derived

Published in:

Foundations of Computer Science, 1990. Proceedings., 31st Annual Symposium on

Date of Conference:

22-24 Oct 1990