Structured Approximation of Toeplitz Matrices and Subspaces

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 $\mathit{T}\in {\mathbb{C}}^{n\times n}$ that satisfies a regularity assumption and is corrupted by an arbitrary $\mathit{E}\in {\mathbb{C}}^{n\times n}$ such that $\mathrm{\parallel }\mathit{E}{\mathrm{\parallel }}_2\le \alpha n$, our algorithm outputs a Toeplitz matrix $\widehat{\mathit{T}}$ of rank exactly $r$ such that $\mathrm{\parallel }\mathit{T}-\widehat{\mathit{T}}{\mathrm{\parallel }}_2\le C\sqrt{r}\mathrm{\parallel }\mathit{E}{\mathrm{\parallel }}_2$, where $C,\alpha >0$ are absolute constants. This performance guarantee is minimax optimal in $n$ and $\mathrm{\parallel }\mathit{E}{\mathrm{\parallel }}_2$. 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.