Uniqueness of the logarithm in the Riemannian quotient manifold of fixed-rank correlation matrices

Determine conditions under which the Riemannian logarithm log_{[X]}([Y]) is unique in the quotient manifold Π^m_k S^{k−1}/O(k), and characterize the set of logarithms when uniqueness fails. Specifically, given [X],[Y]∈Π^m_k S^{k−1}/O(k), ascertain whether the logarithmic map at [X] has a unique preimage of [Y] of minimal norm and, if not, describe the multiplicity and structure of all such logarithms.

Background

In Section 5, the authors develop Riemannian geometric tools on the quotient manifold Π^m_k S^{k−1}/O(k), including geodesics, exponential maps, and a practical procedure to compute a logarithm by aligning points via O(k) and lifting geodesics from the spherical product manifold. They show existence of minimizing geodesics and provide a method to obtain a logarithm but do not analyze its uniqueness.

Understanding the uniqueness of the logarithm is essential for stability of algorithms, well-posedness of inverse problems, and differential calculus on the manifold. The authors explicitly defer this analysis, indicating an unresolved question about when log_{[X]}([Y]) is unique and how to characterize the set of logarithms when it is not.