A geodesic convexity-like structure for the polar decomposition of a square matrix

We make a full landscape analysis of the (generally non-convex) orthogonal Procrustes problem. This problem is equivalent to 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.