Split-Merge: A Difference-based Approach for Dominant Eigenvalue Problem

Abstract: The computation of the dominant eigenvector of symmetric positive semidefinite matrices is a cornerstone operation in numerous optimization-driven applications. Traditional methods, typically based on the \textit{Quotient} formulation, often suffer from challenges related to computational efficiency and reliance on prior spectral knowledge. In this work, we leverage the alternative \textit{Difference} formulation to reinterpret the classical power method as a first-order optimization algorithm. This perspective allows for a novel convergence analysis and facilitates the development of accelerated variants with larger step-sizes, achieving faster convergence without additional computational cost. Building on this insight, we introduce a generalized family of Difference-based methods, with the power method as a special case. Within this family, we propose Split-Merge, an algorithm that attains accelerated convergence without requiring spectral knowledge and operates solely via matrix-vector products. Extensive experiments on both synthetic and real-world datasets demonstrate that Split-Merge consistently outperforms state-of-the-art methods in both efficiency and scalability. In particular, it achieves more than a $\boldsymbol{10\times}$ speedup over the classical power method, underscoring its practical effectiveness for large-scale problems.