Structured Approximation of Toeplitz Matrices and Subspaces

2025-11-21

AI Summary

This paper addresses two structured matrix recovery problems: (1) recovering a low-rank Toeplitz matrix from noisy observations, and (2) reconstructing the column space of a Fourier matrix from a single observation. Both problems are challenging due to the incompatibility of jointly optimizing structural constraints—Toeplitz (or Hankel), low-rankness, and subspace structure. To overcome this, we introduce Gradient-MUSIC—a novel unifying framework integrating spectral estimation with constrained optimization—into structured matrix recovery for the first time. We establish a minimax-optimal error bound under noise: ‖T − T̂‖₂ ≤ C√r ‖E‖₂. Moreover, we provide the first quantitative characterization linking spectral estimation accuracy to structured matrix recovery performance. The method naturally extends to Hankel matrices and exhibits both computational efficiency and strong robustness to noise.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Search and Optimization: Non-convex Optimization
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Security and Privacy: Large-scale security measurements
• Web Mining and Content Analysis: Bridging structured and unstructured data

Abstract

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 ${oldsymbol T}in {mathbb C}^{n imes n}$ that satisfies a regularity assumption and is corrupted by an arbitrary ${oldsymbol E}in {mathbb C}^{n imes n}$ such that $|{oldsymbol E}|_2leq αn$, our algorithm outputs a Toeplitz matrix $widehat{oldsymbol T}$ of rank exactly $r$ such that $|{oldsymbol T}-widehat{oldsymbol T}|_2 leq C sqrt r , |{oldsymbol E}|_2$, where $C,α>0$ are absolute constants. This performance guarantee is minimax optimal in $n$ and $|{oldsymbol E}|_2$. 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.

Problem

• Recovering corrupted low-rank Toeplitz matrices efficiently
• Determining Fourier matrix range from single observation
• Solving structured approximation problems using Gradient-MUSIC

Innovation

• Uses Gradient-MUSIC algorithm for spectral estimation
• Efficiently recovers corrupted low-rank Toeplitz matrices
• Optimally reconstructs Fourier matrix range from observation