A geodesic convexity-like structure for the polar decomposition of a square matrix
Foivos Alimisis, Bart Vandereycken
TL;DR
The paper reveals a geodesic convexity-like (WQSC) structure for the nonconvex orthogonal Procrustes problem, recasting the polar factor extraction as a Riemannian optimization on the orthogonal group $ ext{$ olinebreak$O}(n)$ with the objective $f(X)=-\mathrm{Tr}(C X)$. It derives the exact Riemannian gradient and Hessian, proves weak-quasi-strong-convexity and quadratic growth relative to the Procrustes solution, and shows convergence guarantees for Riemannian gradient descent: linear convergence when $C$ is invertible and an algebraic (or $\mathcal{O}(1/t)$) rate in the singular case. The analysis connects to similar results for symmetric eigenproblems and positions the approach within the broader geometry of the orthogonal group, while suggesting avenues for robust/noisy and Stiefel-manifold extensions. Practically, the work deepens understanding of why the polar factor is computation-friendly and provides rigorous convergence guarantees for a canonical nonconvex optimization on a Lie group.
Abstract
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.