This paper studies two structured approximation problems: (1) Recovering a corrupted low-rank Toeplitz matrix and (2) recovering the range of a Fourier matrix from a single observation. Both problems are computationally challenging because the structural constraints are difficult to enforce directly. We show that both tasks can be solved efficiently and optimally by applying the Gradient-MUSIC algorithm for spectral estimation. For a rank $r$ Toeplitz matrix ${\boldsymbol T}\in {\mathbb C}^{n\times n}$ that satisfies a regularity assumption and is corrupted by an arbitrary ${\boldsymbol E}\in {\mathbb C}^{n\times n}$ such that $\|{\boldsymbol E}\|_2\leq αn$, our algorithm outputs a Toeplitz matrix $\widehat{\boldsymbol T}$ of rank exactly $r$ such that $\|{\boldsymbol T}-\widehat{\boldsymbol T}\|_2 \leq C \sqrt r \, \|{\boldsymbol E}\|_2$, where $C,α>0$ are absolute constants. This performance guarantee is minimax optimal in $n$ and $\|{\boldsymbol E}\|_2$. We derive optimal results for the second problem as well. Our analysis provides quantitative connections between these two problems and spectral estimation. Our results are equally applicable to Hankel matrices with superficial modifications.

翻译：本文研究两个结构化逼近问题：(1) 恢复受损的低秩Toeplitz矩阵；(2) 从单次观测中恢复傅里叶矩阵的值域。由于结构约束难以直接施加，这两个问题在计算上均具有挑战性。我们证明，通过应用谱估计中的Gradient-MUSIC算法，这两个任务均可被高效且最优地求解。对于一个满足正则性假设的秩$r$ Toeplitz矩阵${\boldsymbol T}\in {\mathbb C}^{n\times n}$，其被任意满足$\|{\boldsymbol E}\|_2\leq αn$的${\boldsymbol E}\in {\mathbb C}^{n\times n}$所破坏，我们的算法输出一个秩恰好为$r$的Toeplitz矩阵$\widehat{\boldsymbol T}$，使得$\|{\boldsymbol T}-\widehat{\boldsymbol T}\|_2 \leq C \sqrt r \, \|{\boldsymbol E}\|_2$，其中$C,α>0$为绝对常数。该性能保证在$n$和$\|{\boldsymbol E}\|_2$上是最小最大最优的。我们同样为第二个问题推导出最优结果。我们的分析为这两个问题与谱估计之间建立了定量联系。我们的结果经过表面修改后同样适用于Hankel矩阵。