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.
翻译:更准确地说,我们证明,这些一等矩阵近似问题可以通过最大限度增加特殊理性功能来解决。我们的方法使我们能够表明,关于这两种规范的最佳解决办法的结构完全不同,只有在单值分解已经提供了理想的汉克尔或托普利茨结构的最佳一等近似值时,才会在次要的情况下出现。我们还证明,结构低级近似的卡佐算法总是会与一等的固定点趋同。然而,它通常不会与最佳解决办法趋同,无论是弗罗贝尼乌斯规范还是光谱规范。