Matrix rigidity of random Toeplitz matrices

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).