Optimal Rank-1 Hankel Approximation of Matrices: Frobenius Norm, Spectral Norm and Cadzow's Algorithm
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In this paper we derive optimal rank-1 approximations with Hankel or Toeplitz structure with regard to two different matrix norms, the Frobenius norm and the spectral norm. We show that the optimal solutions with respect to these two norms are usually different 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 show that the often used 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.
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