computational complexity

, Volume 27, Issue 2, pp 305–350 | Cite as

Matrix rigidity of random Toeplitz matrices

  • Oded Goldreich
  • Avishay TalEmail author


A matrix A is said to have rigidity s for rank r if A differs from any matrix of rank r on more than s entries. We prove that random n-by-n Toeplitz matrices over \({\mathbb{F}_{2}}\) (i.e., matrices of the form \({A_{i,j} = a_{i-j}}\) for random bits \({a_{-(n-1)}, \ldots, a_{n-1}}\)) have rigidity \({\Omega(n^3/(r^2\log n))}\) for rank \({r \ge \sqrt{n}}\), with high probability. This improves, for \({r = o(n/\log n \log\log n)}\), over the \({\Omega(\frac{n^2}{r} \cdot\log(\frac{n}{r}))}\) bound that is known for many explicit matrices.

Our result implies that the explicit trilinear \({[n]\times [n] \times [2n]}\) function defined by \({F(x,y,z) = \sum_{i,j}{x_i y_j z_{i+j}}}\) has complexity \({\Omega(n^{3/5})}\) in the multilinear circuit model suggested by Goldreich and Wigderson (Electron Colloq Comput Complex 20:43, 2013), which yields an \({\exp(n^{3/5})}\) lower bound on the size of the so-called canonical depth-three circuits for F. We also prove that F has complexity \({\tilde{\Omega}(n^{2/3})}\) if the multilinear circuits are further restricted to be of depth 2.

In addition, we show that a matrix whose entries are sampled from a \({2^{-n}}\)-biased distribution has complexity \({\tilde{\Omega}(n^{2/3})}\), regardless of depth restrictions, almost matching the known \({O(n^{2/3})}\) upper bound for any matrix. We turn this randomized construction into an explicit 4-linear construction with similar lower bounds, using the quadratic small-biased construction of Mossel et al. (Random Struct Algorithms 29(1):56–81, 2006).


Matrix rigidity multi-linear functions multi-linear circuits 

Subject classification



Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Alon N., Goldreich O., Håstad J., Peralta R. (1992) Simple Construction of Almost k-wise Independent Random Variables. Random Structures and Algorithms 3(3): 289–304MathSciNetCrossRefzbMATHGoogle Scholar
  2. A. E. Andreev (1987). On a method for obtaining more than quadratic effective lower bounds for the complexity of \({\pi}\)-schemes. Moscow Univ. Math. Bull. 42, 63–66. In Russian.Google Scholar
  3. Bürgisser P., Lotz M. (2004) Lower bounds on the bounded coefficient complexity of bilinear maps. J. ACM 51(3): 464–482MathSciNetCrossRefzbMATHGoogle Scholar
  4. Friedman J. (1993) A note on matrix rigidity. Combinatorica 13(2): 235–239MathSciNetCrossRefzbMATHGoogle Scholar
  5. O. Goldreich (2008). Computational Complexity: A Conceptual Perspective. Cambridge University Press.Google Scholar
  6. O. Goldreich & A. Tal (2016). Matrix rigidity of random Toeplitz matrices. In STOC, 91–104.Google Scholar
  7. O. Goldreich & A. Wigderson (2013). On the Size of Depth-Three Boolean Circuits for Computing Multilinear Functions. Electronic Colloquium on Computational Complexity (ECCC) 20, 43.Google Scholar
  8. J. Håstad (1989). Almost Optimal Lower Bounds for Small Depth Circuits. In RANDOMNESS AND COMPUTATION, 6–20. JAI Press.Google Scholar
  9. S. Kopparty, M. Kumar & M. E. Saks (2014). Efficient Indexing of Necklaces and Irreducible Polynomials over Finite Fields. In ICALP, 726–737.Google Scholar
  10. R. Lidl & H. Niederreiter (1997). Finite Fields, volume 20 of Encyclopedia of mathematics and its applications. Cambridge University Press, 2nd edition.Google Scholar
  11. Lokam S.V. (2009) Complexity Lower Bounds using Linear Algebra. Foundations and Trends in Theoretical Computer Science 4(1–2): 1–155MathSciNetzbMATHGoogle Scholar
  12. Mossel E., Shpilka A., Trevisan L. (2006) On epsilon-biased generators in \({NC^{0}}\). Random Structures and Algorithms 29(1): 56–81MathSciNetCrossRefzbMATHGoogle Scholar
  13. Naor J., Naor M. (1993) Small-Bias Probability Spaces: Efficient Constructions and Applications. SIAM J. on Computing 22(4): 838–856MathSciNetCrossRefzbMATHGoogle Scholar
  14. Paturi R., Pudlák P., Saks M.E., Zane F. (2005) An improved exponential-time algorithm for k-SAT. J. ACM 52(3): 337–364MathSciNetCrossRefzbMATHGoogle Scholar
  15. R. Paturi, P. Pudlák & F. Zane (1999). Satisfiability Coding Lemma. Chicago J. Theor. Comput. Sci. 1999.Google Scholar
  16. Raz R. (2003) On the complexity of matrix product. SIAM J. on Computing 32(5): 1356–1369MathSciNetCrossRefzbMATHGoogle Scholar
  17. Shokrollahi M.A., Spielman M.A., Stemann V. (1997) A Remark on Matrix Rigidity. Inf. Process. Lett. 64(6): 283–285MathSciNetCrossRefzbMATHGoogle Scholar
  18. L. G. Valiant (1977). Graph-theoretic arguments in low-level complexity. In Lecture notes in Computer Science, volume 53, 162–176. Springer.Google Scholar

Copyright information

© Springer International Publishing 2016

Authors and Affiliations

  1. 1.Weizmann Institute of ScienceRehovotIsrael
  2. 2.Institute for Advanced StudyPrincetonUSA

Personalised recommendations