Ryoya Fukasaku, Michio Yamamoto, Yutaro Kabata, Yasuhiko Ikematsu, Kei Hirose . Algebraic Approach for Orthomax Rotations . 2025年4月 詳細を見る In exploratory factor analysis, rotation techn2025 · 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. arXiv researchmap その他リンク： http://arxiv.org/pdf/2504.21288v1
Fukasaku R., Hirose K., Kabata Y., Teramoto K. . Algebraic Approach to Maximum Likelihood Factor Analysis . Psychometrika90 ( 5 ) 1 - 592025 · 記述言語： 英語 掲載種別： 研究論文（学術雑誌） 出版者・発行元： Psychometrika In maximum likelihood factor analysis, we need to solve a complicated system of algebraic equations, known as the normal equation, to get maximum likelihood estimates (MLEs). Since this equation is difficult to solve analytically, its solutions are typically computed with continuous optimization methods, such as the Newton–Raphson method. With this procedure, however, the MLEs are dependent on initial values since the log-likelihood function is highly non-concave. Particularly, the estimates of unique variances can result in zero or negative, referred to as improper solutions; in this case, the MLE can be severely unstable. To delve into the issue of the instability, we algebraically compute all candidates for the MLE. We provide an algorithm based on algebraic computations that is carefully designed for maximum likelihood factor analysis. To be specific, Gröbner bases are employed, powerful tools to get simplified sub-problems for given systems of algebraic equations. Our algebraic algorithm provides the MLE independent of the initial values. While computationally demanding, our algebraic approach is applicable to small-scale problems and provides valuable insights into the characterization of improper solutions. For larger-scale problems, we provide numerical methods as practical alternatives to the algebraic approach. We perform numerical experiments to investigate the characteristics of the MLE with our two approaches. DOI： 10.1017/psy.2025.10047 Web of Science Scopus researchmap
Yutaro Kabata, Shigeki Matsutani, Yuta Ogata . Visualization of curvature on curve and surface by tangential angle parametrization . arXiv 2025年8月 詳細を見る We propose a unif2025 · We propose a unified method to visualize curvature on planar curves and surfaces of revolution using the tangential angle parameter. For plane curves, placing markers at equal increments of the tangential angle reveals local bending features and naturally highlights inflection points and vertices. This approach extends to surfaces of revolution, where curvature lines drawn at equal tangential angle steps reflect principal curvature variations and naturally expose ridge and parabolic curves. Our method provides clear, consistent visualizations without arbitrary parameter tuning, offering geometric insight for both analysis and design applications. arXiv researchmap その他リンク： http://arxiv.org/pdf/2508.13708v1
