Matrix rigidity of random Toeplitz matrices

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 𝔽_2 (i.e., matrices of the form A_i,j = a_i-j for random bits a_-(n-1), …, a_n-1 ) have rigidity Ω(n^3/(r^2log n)) for rank r ≥√(n) , with high probability. This improves, for r = o(n/log n loglog n) , over the Ω(n^2/r·log(n/r)) bound that is known for many explicit matrices. Our result implies that the explicit trilinear [n]× [n] × [2n] function defined by F(x,y,z) = ∑_i,jx_i y_j z_i+j has complexity Ω(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 Ω̃(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 Ω̃(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 ).