Generalized Eigenvalue Problem

The generalized eigenvalue problem (GEP) seeks solutions to equations involving two matrices, finding vectors that are scaled differently by each matrix. Current research focuses on developing efficient algorithms, particularly stochastic and gradient-based methods, to solve GEPs, especially in high-dimensional settings and for large datasets, often employing techniques like proximal gradient descent and game-theoretic formulations. These advancements are crucial for various applications, including machine learning (e.g., canonical correlation analysis, feature selection), data analysis (e.g., data collaboration analysis), and even quantum mechanics (solving Schrödinger equations), where efficient GEP solutions improve accuracy and scalability. The development of robust and computationally efficient methods for solving GEPs is driving progress across multiple scientific disciplines.