Optimal Rank-1 Hankel Approximation of Matrices: Frobenius Norm, Spectral Norm, and Cadzow's Algorithm

TL;DR

This paper characterizes optimal rank-1 Hankel and Toeplitz matrix approximations for Frobenius and spectral norms, revealing their distinct structures and analyzing the convergence of Cadzow's algorithm.

Contribution

It introduces a new characterization of rank-1 structured approximations via rational functions and compares their structures under different norms, also analyzing Cadzow's algorithm convergence.

Findings

Optimal solutions differ between Frobenius and spectral norms.

Cadzow's algorithm converges to a fixed point but not necessarily the optimal solution.

Solutions coincide with SVD only in trivial cases.

Abstract

We characterize optimal rank-1 matrix approximations with Hankel or Toeplitz structure with regard to two different norms, the Frobenius norm and the spectral norm, in a new way. More precisely, we show that these rank-1 matrix approximation problems can be solved by maximizing special rational functions. Our approach enables us to show that the optimal solutions with respect to these two norms have completely different structure and only coincide in the trivial case when the singular value decomposition already provides an optimal rank-1 approximation with the desired Hankel or Toeplitz structure. We also prove that the Cadzow algorithm for structured low-rank approximations always converges to a fixed point in the rank-1 case. However, it usually does not converge to the optimal solution, neither with regard to the Frobenius norm nor the spectral norm.