This note demonstrates that we can stably recover all symmetric Toeplitz matrices $\pmb{X}_0\in\mathbb{R}^{n\times n}$ of rank at most $r$ 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 a random matrix with Toeplitz 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, 61(7):4034--4059, 2015).

翻译：本文证明，通过采用核范数最小化方法，我们能够从数量约为$r\log^{2} n$的秩一次高斯测量中稳定恢复所有秩至多为$r$的对称托普利兹矩阵$\pmb{X}_0\in\mathbb{R}^{n\times n}$，且失败概率呈指数衰减。我们的方法利用门德尔森小球法在托普利兹约束下进行下降锥分析。其中的关键要素是确定具有托普利兹结构的随机矩阵的谱范数，这本身可能具有独立的研究价值。这一结果改进了先前的分析，并解决了Chen等人（IEEE信息论汇刊，61(7):4034–4059，2015）提出的猜想。