Given a set of multiple point clouds, how to find the rigid transformations (rotation, reflection, and shifting) such that these point clouds are well aligned? This problem, known as 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 still a challenging computational problem due to the inherent nonconvexity. In this paper, we study the semidefinite programming (SDP) relaxation of the generalized orthogonal Procrustes problems and prove that the tightness of the SDP relaxation holds, i.e., the SDP estimator exactly equals the least squares estimator, if the signal-to-noise ratio (SNR) is relatively large. We also prove that an efficient generalized power method with a proper initialization enjoys global linear convergence to the least squares estimator. In addition, we analyze the Burer-Monteiro factorization and show the corresponding optimization landscape is free of spurious local optima if the SNR is large. This explains why first-order Riemannian gradient methods with random initializations usually produce a satisfactory solution despite the nonconvexity. One highlight of our work is that the theoretical guarantees are purely algebraic and do not require any assumptions on the statistical property of the noise. Our results partially resolve one open problem posed in [Bandeira, Khoo, Singer, 2014] on the tightness of the SDP relaxation in solving the generalized orthogonal Procrustes problem. Numerical simulations are provided to complement our theoretical analysis.

翻译：面对一系列多点云, 如何找到僵硬的转变( 旋转、 反映和移动), 从而让这些点云完全吻合? 这个问题, 被称为全正正正正正正的质质问题( GOPP), 在包括统计、 成像科学和计算机愿景在内的若干科学学科中起着根本作用 。 尽管它具有巨大的实际重要性, 但它仍然是一个具有挑战性的计算问题 。 在本文中, 我们研究半定型程序( SDP), 放松普遍或正正正统的奥氏级低级降压问题, 并证明 SDP 放松的紧凑性, 即, SDP 估测器完全等于最小正正数的标度问题 。 这解释了一个高效的通用电法, 正确初始化问题与最小估量问题相趋近。 此外, 我们分析布列尔- 货币级平级的系数化和显示相应的优化版面面面面面面面平坦度( 如果 SNRI不是最接近的直立度, ), 我们的直立度的定点平正的定度通常需要一个不甚的平正的平比的平比 。