2010/12/03 by Matthias Hein, Hein, Matthias, Thomas Bühler +1 · 11 citations
Computer Science · Engineering · Mathematics · #Advanced Optimization Algorithms Research #Blind Source Separation Techniques #FOS: Computer and information sciences #FOS: Mathematics #Face and Expression Recognition #Machine Learning (cs.LG) #Machine Learning (stat.ML) #Optimization and Control (math.OC) #Sparse and Compressive Sensing Techniques #cs.LG #math.OC #stat.ML
paper · pdf · doi:10.48550/arxiv.1012.0774
Long version of paper accepted at NIPS 2010
arxiv created 2010/12/03 · openalex publication_date 2010/12/03 · arxiv updated 2015/03/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Many problems in machine learning and statistics can be formulated as (generalized) eigenproblems. In terms of the associated optimization problem, computing linear eigenvectors amounts to finding critical points of a quadratic function subject to quadratic constraints. In this paper we show that a certain class of constrained optimization problems with nonquadratic objective and constraints can be understood as nonlinear eigenproblems. We derive a generalization of the inverse power method which is guaranteed to converge to a nonlinear eigenvector. We apply the inverse power method to 1-spectral clustering and sparse PCA which can naturally be formulated as nonlinear eigenproblems. In both applications we achieve state-of-the-art results in terms of solution quality and runtime. Moving beyond the standard eigenproblem should be useful also in many other applications and our inverse power method can be easily adapted to new problems.