2024/11/02 by Zhaoqiang Liu, Wen Li, Liu, Zhaoqiang +3 · 1 citation
Computer Science · Engineering · #FOS: Computer and information sciences #Machine Learning (cs.LG) #Machine Learning (stat.ML) #Matrix Theory and Algorithms #Stability and Control of Uncertain Systems
paper · pdf · doi:10.48550/arxiv.2411.01326
openalex publication_date 2024/11/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Generalized eigenvalue problems (GEPs) find applications in various fields of science and engineering. For example, principal component analysis, Fisher's discriminant analysis, and canonical correlation analysis are specific instances of GEPs and are widely used in statistical data processing. In this work, we study GEPs under generative priors, assuming that the underlying leading generalized eigenvector lies within the range of a Lipschitz continuous generative model. Under appropriate conditions, we show that any optimal solution to the corresponding optimization problems attains the optimal statistical rate. Moreover, from a computational perspective, we propose an iterative algorithm called the Projected Rayleigh Flow Method (PRFM) to approximate the optimal solution. We theoretically demonstrate that under suitable assumptions, PRFM converges linearly to an estimated vector that achieves the optimal statistical rate. Numerical results are provided to demonstrate the effectiveness of the proposed method.