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

TL;DR

This paper explores a convexity-like structure in the non-convex orthogonal Procrustes problem, showing gradient descent efficiently computes the polar factor of a matrix with linear or algebraic convergence.

Contribution

It reveals a convexity-like structure for the polar decomposition problem, explaining its tractability and analyzing gradient descent convergence rates.

Findings

Gradient descent computes the polar factor with linear convergence for invertible matrices.

Gradient descent computes the polar factor with algebraic convergence for singular matrices.

The structure explains the problem's tractability and connects to eigenvalue problem results.

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.