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

TL;DR

This paper proves stable recovery of low-rank symmetric Toeplitz matrices from a near-optimal number of structured measurements using nuclear norm minimization, with a novel descent cone analysis.

Contribution

It introduces a new analysis method employing descent cone analysis and Mendelson's small ball method for Toeplitz matrix recovery, improving previous results.

Findings

Recovery from $r\, ext{log}^2 n$ measurements with high probability

Determines spectral norm of Toeplitz-structured random matrices

Resolves a conjecture from prior work in 2015

Abstract

This note demonstrates that we can stably recover all symmetric Toeplitz matrices $\pmb{X}_0\in\mathbb{R}^{n\times n}$ of rank at most $r$ 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 a random matrix with Toeplitz 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, 61(7):4034--4059, 2015).