The Difference between Optimal Rank‐1 Hankel Approximations in the Frobenius Norm and the Spectral Norm

We present new examples illustrating the fact that optimal Hankel structured rank‐1 approximation of matrices is usually different for the Frobenius and spectral norm. Further, we compare our results for optimal rank‐1 Hankel approximation to the results of the well‐known Cadzow algorithm [3].


Introduction
A Hankel matrix H 1 ∈ C M ×N of rank 1 is either of the form Neglecting the first case, the problem of approximating a given matrix A by a Hankel structured matrix of rank 1 reads min H Hankel rank H=1 In [1] we have solved this problem for both the Frobenius and the spectral norm, where for the spectral norm we restricted our considerations to real symmetric matrices A ∈ R N ×N and real parameters c ∈ R \ {0} and z ∈ R.
Let H 1 =czz T be an optimal rank-1 Hankel approximation of A with regard to the spectral norm.
(i) The optimal error bound A − H 1 and ifc is chosen such that where the primed sum indicates that terms of the form 0 0 might occur for which we use the convention 0 0 = 0.
(ii) If there is noz satisfying (2), then the optimal rank-1 Hankel approximation of A possesses the errorλ :

Examples and Comparisons
In the beginning of this section, we present two examples and the respective optimal rank-1 Hankel approximations. Then, by the means of these examples, we emphasize the difference between the optimal approximation w.r.t. the Frobenius norm and the one w.r.t. the spectral norm. The resulting approximation errors (rounded to four digits) are presented in Table 1.
In this example, the eigenvector corresponding to the largest eigenvalue is of the structure of z from (1) for z = 0. Therefore Cadzow's algorithm (see [1,3]) terminates after one iteration and leads to the optimal result also aquired by Thm. 2.1.
Regarding the optimal approximation w.r.t. the spectral norm, the conditions (2) cannot be satisfied in this example, since v T 1 z = 1 for all z ∈ R. Thus, we apply Thm. 2.2(ii) in form of a bisection iteration and obtainz ≈ 4.1113 andc ≈ 0.0145. Example 3.1 confirmes a theoretical result from [1]: The optimal rank-1 Hankel approximations w.r.t. the Frobenius and spectral norm coincide with the optimal unstructured rank-1 approximation if the eigenvector corresponding to the largest eigenvalue already is of the structure indicated in (1). At the same time we observe that this condition is not necessary for the approximation in the spectral norm to achieve the error bound of the unstructured approximation since (2) is a weaker condition. Furthermore, the approximation in the spectral norm is not unique and it might be a good idea to choosec in (3) such that also the Frobenius norm is minimized.

Alg spectral
In Example 3.2 it is evident that the results for the Frobenius norm and the spectral norm differ significantly as predicted in [1]. Further, this example backs the long-standing suspicion that Cadzow's algorithm in general does not lead to optimal results since for both the Frobenius and the spectral norm we find better approximations.