Solve-Time Anomaly at Intermediate Data Dimensions

Explain why the Riemannian trust-region solver for the complex Normal Procrustes Problem requires substantially more computation time when the number of columns n is near one-half of the number of rows m than when n is near either end of the dimensional range.

Background

The numerical experiments reveal an unexpected nonmonotonic relationship between solve time and the number of columns. Small n produces an effectively underdetermined problem, while n close to m removes much of the unconstrained freedom, but neither observation explains the markedly larger solve times near n=0.5m. The paper explicitly leaves a finer analysis of this behavior unresolved.

References

Why solve times when $n$ is proximal to $0.5m$ are markedly higher than when $n$ is proximal to $m$ remains unclear to the author, and we leave a finer analysis of this tendency to future work.

— The Normal Procrustes Problem: A Riemannian Optimization Approach  (2608.19513 - Bierly, 20 Aug 2026) in Section 4.2, subsection “Optimization Performance on Random Matrices”