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The generalized orthogonal Procrustes problem (GOPP) plays a fundamental rolein several scientific disciplines including statistics, imaging science andcomputer vision. Despite its tremendous practical importance, it is generallyan NP-hard problem to find the least squares estimator. We study thesemidefinite relaxation (SDR) and an iterative method named generalized powermethod (GPM) to find the least squares estimator, and investigate theperformance under a signal-plus-noise model. We show that the SDR recovers theleast squares estimator exactly and moreover the generalized power method witha proper initialization converges linearly to the global minimizer to the SDR,provided that the signal-to-noise ratio is large. The main technique followsfrom showing the nonlinear mapping involved in the GPM is essentially a localcontraction mapping and then applying the well-known Banach fixed-point theoremfinishes the proof. In addition, we analyze the low-rank factorizationalgorithm and show the corresponding optimization landscape is free of spuriouslocal minimizers under nearly identical conditions that enables the success ofSDR approach. The highlight of our work is that the theoretical guarantees arepurely algebraic and do not assume any statistical priors of the additiveadversaries, and thus it applies to various interesting settings.

Generalized Orthogonal Procrustes Problem under Arbitrary Adversaries