Abstract
A threshold gate is a linear function of input variables with integer coefficients (weights). It outputs 1 if the value of the function is positive. The sum of absolute values of coefficients is called the total weight of the threshold gate. A perceptron of order d is a circuit of depth 2 having a threshold gate on the top level and conjunctions of fan-in at most d on the remaining level.
For every n and 
we construct a function computable by a perceptron of order d but not computable by any perceptron of order D with total weight \(2^{o(n^d/D^{4d})}\). In particular, if D is a constant, our function is not computable by any perceptron of order D with total weight \(2^{o(n^d)}\). Previously functions with this properties were known only for d = 1 (and arbitrary D) [2] and for D = d [12].
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
Allender, E.: Circuit complexity before the dawn of the new Millennium. In: Chandru, V., Vinay, V. (eds.) FSTTCS 1996. LNCS, vol. 1180, pp. 1–18. Springer, Heidelberg (1996)
Beigel, R.: Perceptrons, PP and the polynomial hierarchy. Computational Complexity 4, 339–349 (1994)
Beigel, R., Reingold, N., Spielman, D.A.: The perceptron strikes back. In: Proceedings of Structure in Complexity Theory Conference, pp. 286–291 (1991)
Chandra, A., Stockmeyer, L., Vishkin, U.: Constant depth reducibility. SIAM Journal on Computing 13, 423–439 (1984)
Goldmann, M., Håstad, J., Razborov, A.A.: Majority gates vs. general weighted threshold gates. Computational Complexity 2, 277–300 (1992)
Hajnal, A., Maass, W., Pudlák, P., Szegedy, M., Turán, G.: Threshold circuits of bounded depth. Journal on Computer and System Science 46, 129–154 (1993)
Håstad, J.: Almost optimal lower bounds for small depth circuits. In: Micali, S. (ed.) Randomness and Computation, Advances in Computing Research, vol. 5, pp. 143–170. JAI Press Inc. (1989)
Håstad, J.: On the size of weights for threshold gates. SIAM Journal on Discrete Mathematics 7(3), 484–492 (1994)
Minsky, M.L., Papert, S.A.: Perceptrons. MIT Press, Cambridge (1968)
Muroga, S.: Threshold logic and its applications. Wiley-Interscience, Chichester (1971)
Nisan, N.: The communication complexity of threshold gates. In: Miklós, D., Sós, V.T., Szönyi, T. (eds.) Combinatorics, Paul Erdös is Eighty, vol. 1, pp. 301–315. Jason Bolyai Math. Society, Budapest, Hungary (1993)
Podolskii, V.V.: Perceptrons of large weight. In: Proceedings, Second International Symposium on Computer Science in Russia, pp. 328–336 (2007)
Razborov, A.: Lower bounds on the size of bounded-depth networks over a complete basis with logical addition. Matematicheskie Zametki 41(4), 598–607 (1987); English translation in Mathematical Notes of the Academy of Sci. of the USSR 41(4), 333–338 (1987)
Smolensky, R.: Algebraic methods in the theory of lower bounds for Boolean circuit complexity. In: Proceedings, 19th ACM Symposium of Theory of Computing, pp. 77–82 (1987)
Yao, A.: Separating the polynomial-time hierarchy by oracles. In: IEEE Symposium on Foundations of Computer Science (FOCS), pp. 1–10 (1985)
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 2008 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Podolskii, V.V. (2008). A Uniform Lower Bound on Weights of Perceptrons. In: Hirsch, E.A., Razborov, A.A., Semenov, A., Slissenko, A. (eds) Computer Science – Theory and Applications. CSR 2008. Lecture Notes in Computer Science, vol 5010. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-79709-8_27
Download citation
DOI: https://doi.org/10.1007/978-3-540-79709-8_27
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-79708-1
Online ISBN: 978-3-540-79709-8
eBook Packages: Computer ScienceComputer Science (R0)