Convergence of the alternating minimization algorithm for the weighted Fréchet mean in the orbit space

Establish convergence of the iterates (X_s, {O^i_s}) produced by Algorithm 2 (Alternating minimization for the weighted Fréchet mean) when computing the weighted Fréchet mean of points [X^i] in the orbit space Π^m S^{k−1}/O(k) with positive weights w_i. Precisely, prove that the sequence (X_s, {O^i_s}) converges (e.g., to a critical point or to a minimizer) under appropriate conditions, and characterize the nature of the limit points within Π^m S^{k−1}/O(k).

Background

The paper studies the orbit space structure Π^m S^{k−1}/O(k) for correlation matrices of bounded rank and defines a metric d^{•,k} via a quotient construction. Computing weighted Fréchet means in this space reduces to a joint optimization over X in Π^m S^{k−1} and orthogonal alignments {O^i}∈O(k), for which the authors propose an alternating minimization algorithm (Algorithm 2).

They prove the loss function decreases monotonically and converges (Proposition 4.11), but they do not analyze the convergence of the iterates (X_s, {O^i_s}) themselves. A rigorous convergence analysis (e.g., to stationary points or global minimizers, with possible conditions on the data or weights) remains to be established.