Liu Yang
Department of Human Development and Quantitative Methodology, University of Maryland, College Park, MD, USA.
Br J Math Stat Psychol. 2021 Feb;74(1):139-163. doi: 10.1111/bmsp.12211. Epub 2020 Jul 26.
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.
在探索性因子分析中，在进行分析旋转之前，潜在因子和因子载荷很少能够被解释。通常，通过在正交或斜交旋转矩阵的流形中进行数值搜索来解决旋转问题，以使旋转后的因子载荷最小化一个预先指定的复杂度函数。广泛使用的梯度投影（GP）算法虽然编程简单且能够处理正交和斜交旋转，但当显变量数量和/或潜在因子数量较大时，发现其收敛速度较慢。本研究考察了两种黎曼二阶算法在解决旋转问题方面的有效性，这两种算法分别推广了欧几里得空间中成熟的无约束优化的截断牛顿法和信赖域策略。当接近局部最小值时，二阶算法通常超线性甚至二次收敛，比仅线性收敛的一阶算法要好。在蒙特卡罗研究中还进一步观察到，与GP算法相比，黎曼截断牛顿法和信赖域算法不仅需要更少的迭代次数，而且在满足相同收敛标准时所需的处理时间也少得多，尤其是在斜交旋转的情况下。
