Matrix rigidity of random Toeplitz matrices

AbstractA 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 $${\mathbb{F}_{2}}$$F2 (i.e., matrices of the form $${A_{i,j} = a_{i-j}}$$Ai,j=ai-j for random bits $${a_{-(n-1)}, \ldots, a_{n-1}}$$a-(n-1),…,an-1) have rigidity $${\Omega(n^3/(r^2\log n))}$$Ω(n3/(r2logn)) for rank $${r \ge \sqrt{n}}$$r≥n, with high probability. This improves, for $${r = o(n/\log n \log\log n)}$$r=o(n/lognloglogn), over the $${\Omega(\frac{n^2}{r} \cdot\log(\frac{n}{r}))}$$Ω(n2r·log(nr)) bound that is known for many explicit matrices.Our result implies that the explicit trilinear $${[n]\times [n] \times [2n]}$$[n]×[n]×[2n] function defined by $${F(x,y,z) = \sum_{i,j}{x_i y_j z_{i+j}}}$$F(x,y,z)=∑i,jxiyjzi+j has complexity $${\Omega(n^{3/5})}$$Ω(n3/5) in the multilinear circuit model suggested by Goldreich and Wigderson (Electron Colloq Comput Complex 20:43, 2013), which yields an $${\exp(n^{3/5})}$$exp(n3/5) lower bound on the size of the so-called canonical depth-three circuits for F. We also prove that F has complexity $${\tilde{\Omega}(n^{2/3})}$$Ω~(n2/3) 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 $${2^{-n}}$$2-n-biased distribution has complexity $${\tilde{\Omega}(n^{2/3})}$$Ω~(n2/3), regardless of depth restrictions, almost matching the known $${O(n^{2/3})}$$O(n2/3) 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).