Efficacy of Multiple Starts and Improved Initialization

Analyze the efficacy of multiple random starts and improved initialization strategies for avoiding suboptimal stationary points in the real Riemannian optimization method.

Background

The real reduced objective has additional nonconcavity caused by the block-pairing structure, and the numerical experiments show that the solver frequently terminates at suboptimal critical points. The paper identifies improved initialization and multiple starts as possible remedies but explicitly leaves their effectiveness unresolved. Quantifying their ability to improve the probability of reaching a global optimum would address a central practical weakness of the real method.

References

Running more random starts or introducing an improved initialization strategy may increase the likelihood of avoiding such suboptimal stationary points, but we leave an analysis of the efficacy of these methods to future work.

— The Normal Procrustes Problem: A Riemannian Optimization Approach  (2608.19513 - Bierly, 20 Aug 2026) in Section 8.1, subsection “Known Global Optimum Recovery”