Theory of Computing Systems

, Volume 46, Issue 2, pp 301–310 | Cite as

Entropy of Operators or why Matrix Multiplication is Hard for Depth-Two Circuits

Article

Abstract

We consider unbounded fanin depth-2 circuits with arbitrary boolean functions as gates. We define the entropy of an operator f:{0,1}n→{0,1}m as the logarithm of the maximum number of vectors distinguishable by at least one special subfunction of f.

Our main result is that every depth-2 circuit for f requires at least entropy(f) wires. This is reminiscent of a classical lower bound of Nechiporuk on the formula size, and gives an information-theoretic explanation of why some operators require many wires. We use this to prove a tight estimate Θ(n3) of the smallest number of wires in any depth-2 circuit computing the product of two n by n matrices over any finite field. Previously known lower bound for this operator was Ω(n2log n).

Keywords

Boolean circuits Bilinear forms Matrix multiplication Entropy 

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References

  1. 1.
    Alon, N., Karchmer, M., Wigderson, A.: Linear circuits over GF(2). SIAM. J. Comput. 19(6), 1064–1067 (1990) MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Alon, N., Pudlák, P.: Superconcentrators of depth 2 and 3; odd levels help (rarely). J. Comput. Syst. Sci. 48, 194–202 (1994) MATHCrossRefGoogle Scholar
  3. 3.
    Beigel, R., Tarui, J.: On ACC. Comput. Complex. 4, 350–366 (1994) MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Bshouty, N.H.: A lower bound for matrix multiplication. SIAM J. Comput. 18, 759–765 (1982) CrossRefMathSciNetGoogle Scholar
  5. 5.
    Bürgisser, P., Lotz, M.: Lower bounds on the bounded coefficient complexity of bilinear maps. J. ACM 51(3), 464–482 (2004) MathSciNetGoogle Scholar
  6. 6.
    Cherukhin, D.Y.: The lower estimate of complexity in the class of schemes of depth 2 without restrictions on a basis. Moscow University Math. Bull. 60(4), 42–44 (2005) MathSciNetGoogle Scholar
  7. 7.
    Coppersmith, D., Winograd, S.: Matrix multiplications via arithmetic progressions. J. Symb. Comput. 9, 251–280 (1990) MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Dolev, D., Dwork, C., Pippenger, N., Wigderson, A.: Superconcentrators, generalizer and generalized connectors with limited depth. In: Proc. of the 15th STOC, pp. 42–51 (1983) Google Scholar
  9. 9.
    Friedman, J.: A note on matrix rigidity. Combinatorica 13, 235–239 (1993) MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Jukna, S.: On set intersection representations of graphs. J. Graph Theory (submitted) Google Scholar
  11. 11.
    Morgenstern, J.: Note on a lower bound on the linear complexity of fast Fourier transform. J. ACM 20(2), 305–306 (1973) MATHMathSciNetGoogle Scholar
  12. 12.
    Morgenstern, J.: The linear complexity of computation. J. ACM 22(2), 184–194 (1975) MATHGoogle Scholar
  13. 13.
    Nechiporuk, E.I.: On a Boolean function. Sov. Math. Dokl. 7(4), 999–1000 (1966) MATHGoogle Scholar
  14. 14.
    Pippenger, N.: Superconcentrators. SIAM J. Comput. 6, 298–304 (1977) MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Pippenger, N.: Superconcentrators of depth 2. J. Comput. Syst. Sci. 24, 82–90 (1982) MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Pudlák, P.: Communication in bounded depth circuits. Combinatorica 14(2), 203–216 (1994) MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Pudlák, P.: A note on the use of determinant for proving lower bounds on the size of linear circuits. Inf. Process. Lett. 74, 197–201 (2000) CrossRefGoogle Scholar
  18. 18.
    Pudlák, P., Rödl, V.: Some combinatorial-algebraic problems from complexity theory. Discrete Math. 136, 253–279 (1994) MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Pudlák, P., Rödl, V., Sgall, J.: Boolean circuits, tensor ranks, and communication complexity. SIAM J. Comput. 26(3), 605–633 (1997) MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Pudlák, P., Savický, P.: On shifting networks. Theor. Comput. Sci. 116, 415–419 (1993) MATHCrossRefGoogle Scholar
  21. 21.
    Radhakrishnan, J., Ta-Shma, A.: Bounds for dispersers, extractors, and depth-two superconcentrators. SIAM J. Discrete Math. 13(1), 2–24 (2000) MATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Raz, R., Shpilka, A.: Lower bounds for matrix product in bounded depth circuits with arbitrary gates. SIAM J. Comput. 32(2), 488–513 (2003) MATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Shoup, V., Smolensky, R.: Lower bounds for polynomial evaluation and interpolation problems. Comput. Complex. 6(4), 301–311 (1997) CrossRefMathSciNetGoogle Scholar
  24. 24.
    Strassen, V.: Die Berechnungskomplexität von elementarsymmetrischen Funktionen und von Interpoliationskoefizienten. Numer. Math. 20, 238–251 (1973) MATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Valiant, L.: Graph-theoretic methods in low-level complexity. In: Proc. of the 6th MFCS. Lecture Notes in Computer Science, vol. 53, pp. 162–176. Springer, Berlin (1977) Google Scholar
  26. 26.
    Yao, A.C.: On ACC and threshold circuits. In: Proc. of the 31th FOCS, pp. 619–627 (1990) Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  1. 1.Institute of Mathematics and InformaticsVilniusLithuania

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