Uncertainty Quantification of Least Squares Estimator for Generalized Orthogonal Procrustes Problems

The generalized orthogonal Procrustes problem (GOPP) aims to recover rigid transformations that best align multiple point clouds and has broad applications in 3D geometry, computer vision, and biomedicine. Despite its importance, the problem is challenging due to the nonconvex orthogonality constraints. Fortunately, the generalized power method (GPM) has proven highly effective, theoretically guaranteed to converge to the global least squares (LS) estimator under reasonable conditions. Yet, the statistical inference of this global LS estimator remains largely an open problem. Specifically, the absence of an exact second-order analytic expansion prevents rigorous uncertainty quantification (UQ), which is vital for constructing confidence regions and evaluating estimator reliability. While recent UQ frameworks developed for orthogonal group synchronization offer a potential paradigm, they severely lack universality, often failing to accommodate the complex anisotropic distortions inherent in GOPP shape matrices. To bridge this gap, we propose a more general theoretical framework with which we successfully derive the exact second-order analytic expansion of the GOPP global LS estimator under additive Gaussian noise. Leveraging this theoretical foundation, we further construct confidence regions. Finally, extensive numerical experiments empirically validate the exactness of our theoretical expansion and demonstrate the robust coverage of the proposed confidence regions across varying conditions.