Random Toeplitz Matrices: The Condition Number under High Stochastic Dependence

In this paper, we study the condition number of a random Toeplitz matrix. Since a Toeplitz matrix is a diagonal constant matrix, its rows or columns cannot be stochastically independent. This situation does not permit us to use the classic strategy to analyze its minimum singular value when all the entries of a random matrix are stochastically independent. Using a circulant embedding, we can break the stochastic dependence of the structure of the Toeplitz matrix and reduce the problem to analyze the extreme singular values of a random circulant matrix. Among our results, we show the condition number of non--symmetric random circulant matrix of dimension $n$ under the existence of moment generating function of the random entries is $\kappa\left(\mathcal{C}_n\right) = \mbox{O}\left( \frac{1}{\varepsilon} n^{\rho+1/2} \left(\log n\right)^{1/2} \right)$ with probability $1-\mbox{O}\left((\varepsilon^2 + \varepsilon) n^{-2\rho} + n^{-1/2+\scriptstyle{o}(1)}\right)$ for any $\varepsilon >0$, $\rho\in(0,1/4)$. Moreover, if the random entries only have the second moment, we have $\kappa\left(\mathcal{C}_n\right) = \mbox{O}\left( \frac{1}{\varepsilon} n^{\rho+1/2} \log n\right)$ with probability $1-\mbox{O}\left((\varepsilon^2 + \varepsilon) n^{-2\rho} + \left(\log n\right)^{-1/2}\right)$. For the condition number of a random (non--symmetric or symmetric) Toeplitz matrix $\mathcal{T}_n$ we establish $\kappa\left(\mathcal{T}_n\right) \leq \kappa\left(\mathcal{C}_{2n}\right) \left(\sigma_{\min}\left( C_{2n} \right)\sigma_{\min}\left( S_n \right)\right)^{-1}$, where $\sigma_{\min}(A)$ is the minimum singular value of the matrix $A$. The matrix $C_{2n}$ is a random circulant matrix and $S_n:=F^*_{2,n} D_{1,n}^{-1}F_{2,n} + F^*_{4,n} D^{-1}_2 F_{4,n}$, where $F_{2,n},F_{4,n}$ are deterministic matrices and $D_{1,n}, D_{2,n}$ are random diagonal matrices. We conjeture $S_n$ is well conditioned.