Optimal Hankel Structured Rank-1 Approximation

Hankel matrices are closely related to linear time-invariant (LTI) models, which are widely used in areas like system theory, signal processing, computer algebra, or machine learning. The complexity of such a model is related to the rank of this matrix: a simple LTI model corresponds to a Hankel matrix of low rank. Thus, Hankel structured low-rank approximation (SLRA) of a matrix is an important task. The majority of related approaches from the literature only achieves approximate solutions to the SLRA problem with respect to the Frobenius norm. In contrast, for the special case of the rank-1 Hankel approximation (r1H) problem we characterize optimal solutions both for the Frobenius norm and for the spectral norm. More precisely, we show that the r1H problems can be solved by maximizing special rational functions. Since we are able to compute the optimal solutions numerically, they can serve as benchmarks for different methods engaging in the r1H problem. We also give a complete proof that the famous Cadzow algorithm always converges in the r1H setting.