## Matrix Rigidity of Random Toeplitz Matrices

#### Abstract

We prove that random $n$-by-$n$ Toeplitz (alternatively Hankel) matrices over $F_2$ have rigidity $\Omega(\frac{n^3}{r^2\log n})$ 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 (ECCC, 2013), which yields an $\exp(n^{3/5})$ lower bound on the size of the so-called \emph{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 $O(n^{2/3})$ upper bound for any matrix by Goldreich and Wigderson. 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).

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