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.

翻译：广义特征值问题在科学与工程的诸多领域中具有广泛应用。例如，主成分分析、费希尔判别分析以及典型相关分析均为广义特征值问题的具体实例，并广泛运用于统计数据处理。本研究探讨生成先验下的广义特征值问题，假设潜在的领先广义特征向量位于一个利普希茨连续生成模型的取值范围内。在适当条件下，我们证明相应优化问题的任何最优解均能达到最优统计速率。此外，从计算角度出发，我们提出一种名为投影瑞利流法的迭代算法以逼近最优解。我们在理论上证明，在合适的假设下，投影瑞利流法能够线性收敛至一个达到最优统计速率的估计向量。数值实验结果验证了所提方法的有效性。