Matrix rigidity of random Toeplitz matrices

We prove that random n-by-n Toeplitz (alternatively Hankel) matrices over F2 have rigidity Ω(n³/r²log n) for rank r ≥ √n, with high probability. For r = o(n/log n log log n), this improves over the Ω(n²/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) =σ_i,jx_iyjz_i+j has complexity Ω(n^3/5) in the multilinear circuit model suggested by Goldreich and Wigderson (ECCC, 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 Ω∼(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 Ω∼(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. (RS&A, 2006).