AI 中文总结
针对现有文献无正规普罗克拉斯问题求解方法的空白，提出首个基于黎曼流形优化的求解方法，可适配实正规场景，还能用于求解两类最近正规矩阵特例问题，数值实验验证了其残差性能和时间可扩展性。
AI 中文摘要
针对给定的$m \ imes n$数据矩阵$X$、$Y$，我们研究正规普罗克拉斯问题（Normal Procrustes Problem）——这是一个最小化$\|AX-Y\|_F^2$的最小二乘优化问题，其中约束$A$为$m \ imes m$正规矩阵。据本文作者所知，现有文献中尚无其他求解正规普罗克拉斯问题的方法，因此我们提出了据我们所知的首个此类方法。此外，我们调整了该方法以求解实正规普罗克拉斯问题（Real Normal Procrustes Problem），该问题中$A$必须为实矩阵。在处理这些问题时，我们首先将复值和实值目标函数分别简化为可在酉矩阵和实正交矩阵的黎曼流形上直接优化的形式。这种简化使我们能够应用黎曼流形优化技术来逼近两类问题的解。最近正规矩阵问题（Closest Normal Matrix Problem）和实最近正规矩阵问题（Real Closest Normal Matrix Problem）均是对应普罗克拉斯问题的特例，且已有文献对其开展过研究。因此，我们的方法还为这两类问题的近似求解提供了一种全新的黎曼优化方法。我们进一步在所有上述问题上数值测试了方法的性能（包括在最近正规矩阵问题上与已有算法进行对比），结果显示该方法取得了具有竞争力的残差，且在实际运行时间上表现出良好的可扩展性。
英文摘要
For given $m \times n$ data matrices $X, Y$, we investigate the Normal Procrustes Problem---the least squares optimization problem that aims to minimize $\|AX-Y\|_F^2$, where $A$ is constrained to be a normal $m \times m$ matrix. As far as the author of this article is aware, no other method that attempts to solve the Normal Procrustes Problem exists in the literature; we thus propose what is, to our knowledge, the first such method. We, furthermore, adapt our approach to address the Real Normal Procrustes Problem, where $A$ must be real. In our treatment of these problems, we first reduce our complex and real objective functions to be purely optimizable over the Riemannian manifolds of the unitary and real orthogonal matrices, respectively. This reduction enables us to apply techniques in Riemannian manifold optimization to approximate solutions to both. The Closest Normal Matrix and Real Closest Normal Matrix Problems are both special cases of their respective Procrustes Problems and have been previously studied in the literature. Our approach thus recovers a novel Riemannian optimization method for approximating solutions to both these problems. We further numerically test the performance of our method across all such problems (including against previously developed algorithms on the Closest Normal Matrix Problems) and obtain competitive residuals and favorable scaling in wall-clock time.
Comments41 pages, 10 figures. Code available at https://github.com/kbierly/NPP
