In exploratory factor analysis, latent factors and factor loadings are seldom interpretable until analytic rotation is performed. Typically, the rotation problem is solved by numerically searching for an element in the manifold of orthogonal or oblique rotation matrices such that the rotated factor loadings minimize a pre‐specified complexity function. The widely used gradient projection (GP) algorithm, although simple to program and able to deal with both orthogonal and oblique rotation, is found to suffer from slow convergence when the number of manifest variables and/or the number of latent factors is large. The present work examines the effectiveness of two Riemannian second‐order algorithms, which respectively generalize the well‐established truncated Newton and trust‐region strategies for unconstrained optimization in Euclidean spaces, in solving the rotation problem. When approaching a local minimum, the second‐order algorithms usually converge superlinearly or even quadratically, better than first‐order algorithms that only converge linearly. It is further observed in Monte Carlo studies that, compared to the GP algorithm, the Riemannian truncated Newton and trust‐region algorithms require not only much fewer iterations but also much less processing time to meet the same convergence criterion, especially in the case of oblique rotation.

中文翻译：

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探索性因子分析中解析旋转的黎曼牛顿和信任域算法

在探索性因子分析中，在执行分析旋转之前，潜在因子和因子载荷很少可解释。通常，旋转问题是通过在正交或倾斜旋转矩阵的流形中进行数值搜索来解决的，从而使旋转的因子载荷最小化预先指定的复杂度函数。广泛使用的梯度投影 (GP) 算法虽然易于编程并且能够处理正交和倾斜旋转，但发现当显变量和/或潜在因子的数量很大时收敛缓慢。目前的工作检验了两种黎曼二阶算法的有效性，它们分别推广了成熟的截断牛顿和信任域策略，用于在欧几里得空间中进行无约束优化，在解决旋转问题。当接近局部最小值时，二阶算法通常会超线性甚至二次收敛，优于仅线性收敛的一阶算法。在蒙特卡罗研究中进一步观察到，与 GP 算法相比，黎曼截断牛顿算法和信任域算法不仅需要更少的迭代，而且需要更少的处理时间来满足相同的收敛标准，特别是在倾斜的情况下回转。