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

This note demonstrates that we can stably recover rank-<italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">r</i> Toeplitz matrix <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\pmb {X}\in \mathbb {R}^{n\times n}$ </tex-math></inline-formula> from a number of rank-one subgaussian measurements on the order of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$r\log ^{2} n$ </tex-math></inline-formula> 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 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).