The generalized orthogonal Procrustes problem (GOPP) plays a fundamental role in several scientific disciplines including statistics, imaging science and computer vision. Despite its tremendous practical importance, it is generally an NP-hard problem to find the least squares estimator. We study the semidefinite relaxation (SDR) and an iterative method named generalized power method (GPM) to find the least squares estimator, and investigate the performance under a signal-plus-noise model. We show that the SDR recovers the least squares estimator exactly and moreover the generalized power method with a proper initialization converges linearly to the global minimizer to the SDR, provided that the signal-to-noise ratio is large. The main technique follows from showing the nonlinear mapping involved in the GPM is essentially a local contraction mapping and then applying the well-known Banach fixed-point theorem finishes the proof. In addition, we analyze the low-rank factorization algorithm and show the corresponding optimization landscape is free of spurious local minimizers under nearly identical conditions that enables the success of SDR approach. The highlight of our work is that the theoretical guarantees are purely algebraic and do not assume any statistical priors of the additive adversaries, and thus it applies to various interesting settings.

翻译：广义正交普鲁克问题（GOPP）在统计学、成像科学与计算机视觉等多个科学学科中发挥着基础性作用。尽管其具有巨大的实际重要性，但寻找最小二乘估计量通常是一个NP困难问题。我们研究了半定松弛（SDR）和名为广义幂法（GPM）的迭代方法以寻找最小二乘估计量，并探讨了其在信号加噪声模型下的性能。我们证明，当信噪比较大时，SDR能够精确恢复最小二乘估计量，且通过适当初始化的广义幂法线性收敛至SDR的全局最小值。主要技术在于证明GPM中涉及的非线性映射本质上是局部收缩映射，随后应用著名的巴拿赫不动点定理完成证明。此外，我们分析了低秩分解算法，并证明在几乎相同条件下对应的优化景观不存在虚假局部最小值，这确保了SDR方法的成功。我们工作的亮点在于理论保证是纯代数的，不假设加性对手的任何统计先验，因此适用于各种有趣的场景。