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).

中文翻译：

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随机 Toeplitz 矩阵的矩阵刚性

摘要如果矩阵 A 与任何秩为 r 的矩阵在 s 个以上的条目上不同，则称矩阵 A 对秩为 r 具有刚性 s。我们证明了 $${\mathbb{F}_{2}}$$F2 上的随机 n×n Toeplitz 矩阵（即 $${A_{i,j} = a_{ij}} $$Ai,j=ai-j 随机位 $${a_{-(n-1)}, \ldots, a_{n-1}}$$a-(n-1),...,an-1 ) 具有刚性 $${\Omega(n^3/(r^2\log n))}$$Ω(n3/(r2logn)) 等级 $${r \ge \sqrt{n}}$$r ≥n，概率高。对于 $${r = o(n/\log n \log\log n)}$$r=o(n/lognloglogn)，这比 $${\Omega(\frac{n^2}{ r} \cdot\log(\frac{n}{r}))}$$Ω(n2r·log(nr)) 界，对于许多显式矩阵是已知的。我们的结果意味着显式三线性 $${[n ]\times [n] \times [2n]}$$[n]×[n]×[2n] 函数定义为 $${F(x,y,z) = \sum_{i,j}{x_i y_j z_{i+j}}}$$F(x,y,z)=∑i, jxiyjzi+j 在 Goldreich 和 Wigderson 建议的多线性电路模型中具有复杂性 $${\Omega(n^{3/5})}$$Ω(n3/5) (Electron Colloq Comput Complex 20:43, 2013)，这产生了 F 的所谓规范深度三回路大小的 $${\exp(n^{3/5})}$$exp(n3/5) 下界。我们还证明 F 有复杂性 $${\tilde{\Omega}(n^{2/3})}$$Ω~(n2/3) 如果多线性电路进一步限制为深度 2。此外，我们证明矩阵其条目是从 $${2^{-n}}$$2-n-biased 分布中采样的具有复杂性 $${\tilde{\Omega}(n^{2/3})}$$Ω~(n2 /3)，不管深度限制如何，几乎匹配任何矩阵的已知 $${O(n^{2/3})}$$O(n2/3) 上限。我们将这个随机构造变成一个显式的 4-线性构造，具有相似的下界，使用 Mossel 等人的二次小偏置构造。（随机结构算法 29(1):56–81, 2006）。