This note demonstrates that we can stably recover rank $r$ Toeplitz matrix $\pmb{X}\in\mathbb{R}^{n\times n}$ from a number of rank one subgaussian measurements on the order of $r\log^{2} n$ with an exponentially decreasing failure probability by employing a nuclear norm minimization program. Our approach utilizes descent cone analysis through Mendelson's small ball method with the Toeplitz constraint. The key ingredient is to determine the spectral norm of the random matrix of the Topelitz structure, which may be of independent interest.This improves upon earlier analyses and resolves the conjecture in Chen et al. (IEEE Transactions on Information Theory, 2015).

翻译：本文证明，通过采用核范数最小化程序，我们可以从数量级为$r\log^{2} n$的秩一次高斯测量中稳定地恢复秩$r$的Toeplitz矩阵$\pmb{X}\in\mathbb{R}^{n\times n}$，且失败概率呈指数衰减。我们的方法利用Mendelson小球的下降锥分析，并引入Toeplitz约束。关键步骤在于确定具有Toeplitz结构的随机矩阵的谱范数，这一结果可能具有独立的研究价值。该结论改进了先前的分析，并解决了Chen等人（IEEE Transactions on Information Theory, 2015）中的猜想。