论文标题
最佳级别矩阵的最佳等级-1：Frobenius Norm，Spectral Norm和Cadzow算法
Optimal Rank-1 Hankel Approximation of Matrices: Frobenius Norm, Spectral Norm, and Cadzow's Algorithm
论文作者
论文摘要
我们以两种不同的规范（Frobenius Norms and Spectral Norm）以新的方式表征了具有Hankel或Toeplitz结构的最佳级别-1矩阵近似值。更确切地说，我们表明这些等级-1矩阵近似问题可以通过最大化特殊有理功能来解决。我们的方法使我们能够证明，相对于这两个规范的最佳解决方案具有完全不同的结构，并且仅在奇异值分解已经提供了所需的Hankel或Toeplitz结构的最佳秩-1近似值时，仅在微不足道的情况下重合。 我们还证明，用于结构化低级别近似值的Cadzow算法总是会收敛到等级-1情况下的固定点。但是，对于Frobenius Norm和光谱规范，它通常不会收敛到最佳解决方案。
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.
