We make a full landscape analysis of the (generally non-convex) orthogonal Procrustes problem. This problem is equivalent with computing the polar factor of a square matrix. We reveal a convexity-like structure, which explains the already established tractability of the problem and show that gradient descent in the orthogonal group computes the polar factor of a square matrix with linear convergence rate if the matrix is invertible and with an algebraic one if the matrix is singular. These results are similar to the ones of Alimisis and Vandereycken (2024) for the symmetric eigenvalue problem. We present an instance of a distributed Procrustes problem, which is hard to deal by standard techniques from numerical linear algebra. Our theory though can provide a solution.

翻译：我们对（通常非凸的）正交Procrustes问题进行了完整的全景分析。该问题等价于计算方阵的极因子。我们揭示了一种类凸性结构，这解释了该问题已有的易处理性，并证明在正交群上的梯度下降能以线性收敛速率计算可逆方阵的极因子，对于奇异矩阵则具有代数收敛速率。这些结果与Alimisis和Vandereycken（2024）关于对称特征值问题的研究结论相似。我们提出了一个分布式Procrustes问题的实例，该问题难以通过数值线性代数的标准技术处理，但我们的理论能够为此提供解决方案。