Generalized Eigenvalue Problems with Generative Priors
This work addresses GEPs with generative priors for applications like PCA and discriminant analysis, offering a novel computational approach with theoretical guarantees, though it appears incremental as it builds on existing GEP frameworks.
The paper tackles generalized eigenvalue problems (GEPs) by assuming the leading eigenvector lies within a generative model's range, showing that optimal solutions achieve the optimal statistical rate and proposing the Projected Rayleigh Flow Method (PRFM) algorithm, which converges linearly and demonstrates effectiveness in numerical results.
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.