Title: Algebraic Approach for Orthomax Rotations Show Chinese title

Title: 代数方法用于正交旋转

Abstract: In exploratory factor analysis, rotation techniques are employed to derive interpretable factor loading matrices. Factor rotations deal with equality-constrained optimization problems aimed at determining a loading matrix based on measure of simplicity, such as ``perfect simple structure'' and ``Thurstone simple structure.'' Numerous criteria have been proposed, since the concept of simple structure is fundamentally ambiguous and involves multiple distinct aspects. However, most rotation criteria may fail to consistently yield a simple structure that is optimal for analytical purposes, primarily due to two challenges. First, existing optimization techniques, including the gradient projection descent method, exhibit strong dependence on initial values and frequently become trapped in suboptimal local optima. Second, multifaceted nature of simple structure complicates the ability of any single criterion to ensure interpretability across all aspects. In certain cases, even when a global optimum is achieved, other rotations may exhibit simpler structures in specific aspects. To address these issues, obtaining all equality-constrained stationary points -- including both global and local optima -- is advantageous. Fortunately, many rotation criteria are expressed as algebraic functions, and the constraints in the optimization problems in factor rotations are formulated as algebraic equations. Therefore, we can employ computational algebra techniques that utilize operations within polynomial rings to derive exact all equality-constrained stationary points. Unlike existing optimization methods, the computational algebraic approach can determine global optima and all stationary points, independent of initial values. We conduct Monte Carlo simulations to examine the properties of the orthomax rotation criteria, which generalizes various orthogonal rotation methods. Show Chinese abstract

Abstract: 在探索性因素分析中，旋转技术被用来得出可解释的因素负荷矩阵。因子旋转处理的是旨在根据简单性度量（如“完全简单结构”和“瑟斯顿简单结构”）确定负荷矩阵的等式约束优化问题。自简单结构的概念本质上具有模糊性，并涉及多个不同的方面以来，已经提出了许多标准。然而，大多数旋转标准可能无法始终如一地产生在分析目的上最优的简单结构，这主要是由于两个挑战。首先，现有的优化技术，包括梯度投影下降法，表现出对初始值的强烈依赖性，并且经常陷入次优的局部最优解。其次，简单结构的多面性使得任何单一标准都难以确保在所有方面都具有可解释性。在某些情况下，即使达到全局最优解，其他旋转在特定方面可能表现出更简单的结构。为了解决这些问题，获得所有等式约束的稳定点——包括全局和局部最优解是有利的。幸运的是，许多旋转标准被表达为代数函数，并且因子旋转中的约束被公式化为代数方程。因此，我们可以采用利用多项式环内运算的计算代数技术来得出所有等式约束的稳定点。与现有的优化方法不同，计算代数方法可以独立于初始值确定全局最优解和所有稳定点。我们进行了蒙特卡洛模拟来检查正交最大旋转标准的性质，该标准概括了各种正交旋转方法。