Abstract
The importance of using the “right” representation of a function in order to “approximate” it has been widely recognized. The Fourier Transform representation of a function is a classic representation which is widely used to approximate real functions (i.e. functions whose inputs are real numbers). However, the Fourier Transform representation for functions whose inputs are boolean has been far less studied. On the other hand it seems that the Fourier Transform representation can be used to learn many classes of boolean functions.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
M. Ajtai. formulae on finite structure. Annals of Pure and Applied Logic, 24:1–48, 1983.
William Aiello and Milena Mihail. Learning the fourier spectrum of probabilistic lists and trees. In Proceedings SODA 91, pages 291–299. ACM, Jan 1991.
Mihir Bellare. A technique for upper bounding the spectral norm with applications to learning. In 5th Annual Workshop on Computational Learning Theory, pages 62–70, July 1992.
Y. Brandman, J. Hennessy, and A. Orlitsky. A spectral lower bound technique for the size of decision trees and two level circuits. IEEE Trans. on Computers., 39(2):282–287, 1990.
J. Bruck. Harmonic analysis of polynomial threshold functions. Siam J. on Disc. Math., 3(2): 168–177, May 1990.
J. Bruck and R. Smolensky. Polynomial threshold functions, AC° functions and spectral norms. In 31th Annual Symposium on Foundations of Computer Science, St. Louis, Missouri, pages 632–641, October 1990.
Merrick L. Furst, Jeffrey C. Jackson, and Sean W. Smith. Improved learning of AC° functions. In 4th Annual Workshop on Computational Learning Theory, pages 317–325, August 1991.
M. Furst, J. Saxe, and M. Sipser. Parity, circuits, and the polynomial time hierarchy. Mathematical Systems Theory, 17:13–27, 1984.
O. Goldreich and L. Levin. A hard-core predicate for all one-way functions. In Proc. 21st ACM Symposium on Theory of Computing, pages 25–32. ACM, 1989.
J. Hastad. Computational limitations for small depth circuits. MIT Press, 1986. Ph.D. thesis.
Torben Hagerup and Christine Rub. A guided tour to chernoff bounds. Info. Proc. Lett, 33:305–308, 1989.
Michael Kharitonov. Cryptographic hardness of distribution-specific learning. In Proceedings of STOC’ 93, pages 372–381. ACM, 1993.
J. Kahn, G. Kalai, and N. Linial. The influence of variables on boolean functions. In 29 th Annual Symposium on Foundations of Computer Science, White Plains, New York, pages 68–80, October 1988.
E. Kushilevitz and Y. Mansour. Learning decision trees using the fourier spectrum. In Proceedings of the 23 rd Annual ACM Symposium on Theory of Computing, pages 455–464, May 1991. (To appear in Siam J. on Computing
N. Linial, Y. Mansour, and N. Nisan. Constant depth circuits, Fourier Transform and learnability. In 30 th Annual Symposium on Foundations of Computer Science, Research Triangle Park, NC, pages 574–579, October 1989.
Yishay Mansour. An O(n log n) learning algorithm for DNF under the uniform distribution. In Workshop on Computational Learning Theory, pages 53–61, July 1992.
Ronald L. Rivest. Learning decision lists. Machine Learning, 2(3):229–246, 1987.
Kai-Yeung Siu and Jehoshua Bruck. On the power of threshold circuits with small weights. Siam J. on Disc. Math., 4(3):423–435, Aug 1991.
A. C. Yao. Separating the polynomial-time hierarchy by oracles. In 26th Annual Symposium on Foundations of Computer Science, Portland, Oregon, pages 1–10, October 1985.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1994 Springer Science+Business Media New York
About this chapter
Cite this chapter
Mansour, Y. (1994). Learning Boolean Functions via the Fourier Transform. In: Roychowdhury, V., Siu, KY., Orlitsky, A. (eds) Theoretical Advances in Neural Computation and Learning. Springer, Boston, MA. https://doi.org/10.1007/978-1-4615-2696-4_11
Download citation
DOI: https://doi.org/10.1007/978-1-4615-2696-4_11
Publisher Name: Springer, Boston, MA
Print ISBN: 978-1-4613-6160-2
Online ISBN: 978-1-4615-2696-4
eBook Packages: Springer Book Archive