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- W2405242020 abstract "A 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 $${mathbb{F}_{2}}$$ (i.e., matrices of the form $${A_{i,j} = a_{i-j}}$$ for random bits $${a_{-(n-1)}, ldots, a_{n-1}}$$ ) have rigidity $${Omega(n^3/(r^2log n))}$$ for rank $${r ge sqrt{n}}$$ , with high probability. This improves, for $${r = o(n/log n loglog n)}$$ , over the $${Omega(frac{n^2}{r}cdotlog(frac{n}{r}))}$$ 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 (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 $${tilde{Omega}(n^{2/3})}$$ 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 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)." @default.
- W2405242020 created "2016-06-24" @default.
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- W2405242020 date "2015-01-01" @default.
- W2405242020 modified "2023-09-26" @default.
- W2405242020 title "Matrix Rigidity of Random Toeplitz Matrices." @default.
- W2405242020 hasPublicationYear "2015" @default.
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