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Structured Approximation of Toeplitz Matrices and Subspaces

Albert Fannjiang, Weilin Li

TL;DR

The paper presents a unified framework that maps two challenging structured approximation problems—low-rank Toeplitz matrix recovery and Fourier subspace estimation—onto spectral estimation via Gradient-MUSIC. It proves minimax-optimal recovery guarantees: a rank-$r$ Toeplitz matrix can be recovered with spectral-norm error bounded by $C\sqrt{r}\,\|{\boldsymbol E}\|_2$, and a Fourier subspace can be recovered with a bound on the subspace distance that scales as $\sqrt{r/n}\,\|{\boldsymbol z}\|_2$, under natural regularity and noise assumptions. The approach yields efficient algorithms, guarantees, and a Hankel extension, with a demonstrated link between spectral estimation and structured matrix problems. The results suggest a transference principle: optimal spectral estimation informs optimal structured approximation, with practical impact for imaging, deconvolution, and related signal-processing tasks.

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 ${\boldsymbol T}\in {\mathbb C}^{n\times n}$ 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.

Structured Approximation of Toeplitz Matrices and Subspaces

TL;DR

The paper presents a unified framework that maps two challenging structured approximation problems—low-rank Toeplitz matrix recovery and Fourier subspace estimation—onto spectral estimation via Gradient-MUSIC. It proves minimax-optimal recovery guarantees: a rank- Toeplitz matrix can be recovered with spectral-norm error bounded by , and a Fourier subspace can be recovered with a bound on the subspace distance that scales as , under natural regularity and noise assumptions. The approach yields efficient algorithms, guarantees, and a Hankel extension, with a demonstrated link between spectral estimation and structured matrix problems. The results suggest a transference principle: optimal spectral estimation informs optimal structured approximation, with practical impact for imaging, deconvolution, and related signal-processing tasks.

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 Toeplitz matrix that satisfies a regularity assumption and is corrupted by an arbitrary such that , our algorithm outputs a Toeplitz matrix of rank exactly such that , where are absolute constants. This performance guarantee is minimax optimal in and . 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.

Paper Structure

This paper contains 18 sections, 9 theorems, 82 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Motivation
  3. Main results
  4. Brief comparison with prior work
  5. Organization
  6. Notation
  7. Review of spectral estimation and Gradient-MUSIC
  8. Approximation by low-rank Toeplitz matrices
  9. Problem formulation
  10. Gradient-MUSIC Toeplitz estimator
  11. Lower bound and optimality
  12. Fourier subspace estimation
  13. Problem formulation
  14. Gradient-MUSIC for Fourier subspace estimation
  15. Lower bound and optimality
  16. ...and 3 more sections

Key Result

Theorem 2.1

Let $n\geq 100$. Let ${\boldsymbol x}\subseteq \mathbb{T}$ such that $\Delta({\boldsymbol x})\geq 8\pi/n$ and ${\boldsymbol U}={\rm range}({\boldsymbol \Phi}(n,{\boldsymbol x}))$ be its associated Fourier subspace in $\mathbb{C}^n$. For any subspace ${\boldsymbol W}$ of the same dimension as ${\bold

Theorems & Definitions (18)

  • Theorem 2.1
  • Lemma 2.2: fannjiang2025optimality Lemma 4.3
  • Theorem 3.1
  • proof
  • Theorem 3.2
  • proof
  • Corollary 3.3
  • proof
  • Remark 3.4
  • Lemma 4.1
  • ...and 8 more