Algebraic Approach for Orthomax Rotations

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.