Under mild conditions on the noise level of the measurements, rotation averaging satisfies strong duality, which enables global solutions to be obtained via semidefinite programming (SDP) relaxation. However, generic solvers for SDP are rather slow in practice, even on rotation averaging instances of moderate size, thus developing specialised algorithms is vital. In this paper, we present a fast algorithm that achieves global optimality called rotation coordinate descent (RCD). Unlike block coordinate descent (BCD) which solves SDP by updating the semidefinite matrix in a row-by-row fashion, RCD directly maintains and updates all valid rotations throughout the iterations. This obviates the need to store a large dense semidefinite matrix. We mathematically prove the convergence of our algorithm and empirically show its superior efficiency over state-of-the-art global methods on a variety of problem configurations. Maintaining valid rotations also facilitates incorporating local optimisation routines for further speed-ups. Moreover, our algorithm is simple to implement; see supplementary material for a demonstration program.
翻译:在测量的噪音水平的温和条件下,平均轮换平均满足强烈的双重性,通过半限定程序(SDP)的放松可以获得全球解决方案。然而,SDP的通用解决方案在实践中相当缓慢,即使是中等规模的旋转平均情况也是如此,因此开发专业化的算法至关重要。在本文中,我们提出了一个快速算法,实现全球最佳的轮换协调下降(RCD),而区块协调下降(BCD)则通过逐行更新半限定矩阵解决SDP,刚果民盟直接维持并更新整个迭代的所有有效轮换。这避免了储存一个大密度的半限定矩阵的需要。我们从数学上证明我们算法的趋同和实验性地表明,它比各种问题配置的最新全球方法更具有优越的效率。保持有效的轮换还有助于将地方优化常规纳入进一步加速。此外,我们的算法简单易执行;见示范方案的补充材料。