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Low-Rank Toeplitz Matrix Restoration: Descent Cone Analysis and Structured Random Matrix

Gao Huang, Song Li

TL;DR

The paper addresses stable recovery of a low-rank Toeplitz matrix $X_0$ from rank-one subgaussian measurements $b_k=\langle \xi_k\xi_k^T, X_0\rangle+e_k$ under a Toeplitz constraint, via nuclear-norm minimization. It develops a descent-cone framework combined with Mendelson's small-ball method restricted to Toeplitz matrices, avoiding stochastic RIP arguments. A key contribution is establishing a near-optimal sample complexity $m \ge L r \log^2 n$ with exponentially decaying failure probability, together with a spectral-norm bound for Toeplitz-structured random matrices obtained through a circulant-embedding technique. The results hold for all subgaussian moments $\mu \ge 1$ and $\ell_p$-bounded noise ($p \ge 1$), broadening applicability beyond previous work and resolving a conjecture in the literature.

Abstract

This note demonstrates that we can stably recover rank $r$ Toeplitz matrix $\pmb{X}\in\mathbb{R}^{n\times n}$ 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 the random matrix of the Topelitz 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, 2015).

Low-Rank Toeplitz Matrix Restoration: Descent Cone Analysis and Structured Random Matrix

TL;DR

The paper addresses stable recovery of a low-rank Toeplitz matrix from rank-one subgaussian measurements under a Toeplitz constraint, via nuclear-norm minimization. It develops a descent-cone framework combined with Mendelson's small-ball method restricted to Toeplitz matrices, avoiding stochastic RIP arguments. A key contribution is establishing a near-optimal sample complexity with exponentially decaying failure probability, together with a spectral-norm bound for Toeplitz-structured random matrices obtained through a circulant-embedding technique. The results hold for all subgaussian moments and -bounded noise (), broadening applicability beyond previous work and resolving a conjecture in the literature.

Abstract

This note demonstrates that we can stably recover rank Toeplitz matrix from a number of rank one subgaussian measurements on the order of 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 the random matrix of the Topelitz 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, 2015).
Paper Structure (6 sections, 6 theorems, 50 equations)

This paper contains 6 sections, 6 theorems, 50 equations.

Table of Contents

  1. Introduction
  2. Proofs
  3. Preliminaries
  4. Descent Cone Analysis
  5. Structured Random Matrix
  6. Proof of Theorem \ref{['thm']}

Key Result

Theorem 1

Let $p\ge1$ and $\left\lVert\pmb{e}\right\lVert_{\ell_p}\le \eta$. Suppose the sample matrices $\left \{\pmb{\xi}_{k}\pmb{\xi}^{\mathsf{T}}_{k} \right\}_{k=1}^{m}$ satisfy $\pmb{\xi}_k$ are given as independent copies of a random vector $\pmb{\xi}$, whose entries $\xi$ are assumed to be i.i.d. with holds simultaneously for all symmetric Toeplitz matrices $\pmb{X}_0$ of rank at most $r$, provided

Theorems & Definitions (11)

  • Theorem 1
  • Remark 1
  • Lemma 1
  • Lemma 2
  • Definition 1
  • Proposition 1
  • proof
  • Theorem 2
  • proof
  • Theorem 3
  • ...and 1 more