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computational complexity

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

Matrix rigidity of random Toeplitz matrices

  • Oded Goldreich
  • Avishay TalEmail author
Article
  • 87 Downloads

Abstract

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).

Keywords

Matrix rigidity multi-linear functions multi-linear circuits 

Subject classification

68Q17 

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Copyright information

© Springer International Publishing 2016

Authors and Affiliations

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

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