在生成模型先验下求解广义特征值问题，提升数据降维与分析精度。
Generalized Eigenvalue Problems with Generative Priors
- 假设主广义特征向量来自光滑生成模型，构建优化框架。
- 所提算法线性收敛，达到最优统计估计率。
- 适用于高维数据分析，尤其适合生成模型驱动场景。
广义特征值问题（GEPs）在科学与工程中应用广泛，如主成分分析、Fisher判别分析和典型相关分析均属其特例，广泛用于统计数据分析。本文研究在生成先验下的GEPs，假设主导广义特征向量位于Lipschitz连续生成模型的像空间中。在适当条件下，证明任意优化问题的最优解能达到最优统计速率。从计算角度，提出一种迭代算法——投影瑞利流方法（PRFM），理论上证明在合理假设下，该方法线性收敛至达到最优统计速率的估计向量。数值实验验证了所提方法的有效性。
原文摘要 · Abstract (English)
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.
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