# Generalized Eigenvalue Problems with Generative Priors

arXiv: https://arxiv.org/abs/2411.01326 · HTML: https://mlsanity.com/paper/2411.01326/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 34 of its 97 references that are in the map.
- First posted on arXiv 2 Nov 2024.

## Builds on (most cited first, 30 of 34)

- [Deep Learning Face Attributes in the Wild](https://mlsanity.com/paper/1411.7766/) (2014)
- [Compressed Sensing using Generative Models](https://mlsanity.com/paper/1703.03208/) (2017)
- [Deep Learning Techniques for Inverse Problems in Imaging](https://mlsanity.com/paper/2005.06001/) (2020)
- [Deep Decoder: Concise Image Representations from Untrained Non-convolutional Networks](https://mlsanity.com/paper/1810.03982/) (2018)
- [Phase Retrieval Under a Generative Prior](https://mlsanity.com/paper/1807.04261/) (2018)
- [Solving Linear Inverse Problems Using GAN Priors: An Algorithm with Provable Guarantees](https://mlsanity.com/paper/1802.08406/) (2018)
- [Compressed Sensing with Deep Image Prior and Learned Regularization](https://mlsanity.com/paper/1806.06438/) (2018)
- [Eigenvalue and Generalized Eigenvalue Problems: Tutorial](https://mlsanity.com/paper/1903.11240/) (2019)
- [Instance-Optimal Compressed Sensing via Posterior Sampling](https://mlsanity.com/paper/2106.11438/) (2021)
- [Robust Compressed Sensing using Generative Models](https://mlsanity.com/paper/2006.09461/) (2020)
- [Algorithmic Guarantees for Inverse Imaging with Untrained Network Priors](https://mlsanity.com/paper/1906.08763/) (2019)
- [Statistical and computational trade-offs in estimation of sparse principal components](https://mlsanity.com/paper/1408.5369/) (2014)
- [Sparse CCA: Adaptive Estimation and Computational Barriers](https://mlsanity.com/paper/1409.8565/) (2014)
- [Sample Complexity Bounds for 1-bit Compressive Sensing and Binary Stable Embeddings with Generative Priors](https://mlsanity.com/paper/2002.01697/) (2020)
- [The spiked matrix model with generative priors](https://mlsanity.com/paper/1905.12385/) (2019)
- [Exact asymptotics for phase retrieval and compressed sensing with random generative priors](https://mlsanity.com/paper/1912.02008/) (2019)
- [Alternating Phase Projected Gradient Descent with Generative Priors for Solving Compressive Phase Retrieval](https://mlsanity.com/paper/1903.02707/) (2019)
- [The Generalized Lasso with Nonlinear Observations and Generative Priors](https://mlsanity.com/paper/2006.12415/) (2020)
- [Provable Compressed Sensing with Generative Priors via Langevin Dynamics](https://mlsanity.com/paper/2102.12643/) (2021)
- [Towards Sample-Optimal Compressive Phase Retrieval with Sparse and Generative Priors](https://mlsanity.com/paper/2106.15358/) (2021)
- [Generative Principal Component Analysis](https://mlsanity.com/paper/2203.09693/) (2022)
- [Sparse PCA through Low-rank Approximations](https://mlsanity.com/paper/1303.0551/) (2013)
- [Efficient Algorithms for Large-scale Generalized Eigenvector Computation and Canonical Correlation Analysis](https://mlsanity.com/paper/1604.03930/) (2016)
- [Sparse Generalized Eigenvalue Problem with Application to Canonical Correlation Analysis for Integrative Analysis of Methylation and Gene Expression Data](https://mlsanity.com/paper/1611.01066/) (2016)
- [Misspecified Phase Retrieval with Generative Priors](https://mlsanity.com/paper/2210.05571/) (2022)
- [A Unified Framework for Uniform Signal Recovery in Nonlinear Generative Compressed Sensing](https://mlsanity.com/paper/2310.03758/) (2023)
- [Sparse Generalized Eigenvalue Problem via Smooth Optimization](https://mlsanity.com/paper/1408.6686/) (2014)
- [A Convex Sparse PCA for Feature Analysis](https://mlsanity.com/paper/1411.6233/) (2014)
- [Sparse Generalized Eigenvalue Problem: Optimal Statistical Rates via Truncated Rayleigh Flow](https://mlsanity.com/paper/1604.08697/) (2016)
- [Optimal Feature Selection in High-Dimensional Discriminant Analysis](https://mlsanity.com/paper/1306.6557/) (2013)

## Built on by (most cited first, 3 of 3)

- [Integrating Intermediate Layer Optimization and Projected Gradient Descent for Solving Inverse Problems with Diffusion Models](https://mlsanity.com/paper/2505.20789/) (2025)
- [Learning Single Index Models with Diffusion Priors](https://mlsanity.com/paper/2505.21135/) (2025)
- [Diffusion Model Based Signal Recovery Under 1-Bit Quantization](https://mlsanity.com/paper/2511.12471/) (2025)

## Abstract (arXiv)

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.