mlsanityData to 7 Oct 2026

Generative Principal Component Analysis

arXiv 2203.09693 ↗First posted 18 Mar 2022stat.ML cs.IT cs.LGcited by 12

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Abstract (arXiv)

In this paper, we study the problem of principal component analysis with generative modeling assumptions, adopting a general model for the observed matrix that encompasses notable special cases, including spiked matrix recovery and phase retrieval. The key assumption is that the underlying signal lies near the range of an L-Lipschitz continuous generative model with bounded k-dimensional inputs. We propose a quadratic estimator, and show that it enjoys a statistical rate of order √fracklog Lm, where m is the number of samples. We also provide a near-matching algorithm-independent lower bound. Moreover, we provide a variant of the classic power method, which projects the calculated data onto the range of the generative model during each iteration. We show that under suitable conditions, this method converges exponentially fast to a point achieving the above-mentioned statistical rate. We perform experiments on various image datasets for spiked matrix and phase retrieval models, and illustrate performance gains of our method to the classic power method and the truncated power method devised for sparse principal component analysis.

Key facts. Cited by 12 papers in the map (data through 7 Oct 2026). Builds on 33 of its 79 references that are in the map. First posted on arXiv 18 Mar 2022.

Builds on

33 of 79 references.

References · 33

  1. 2017Fashion-MNIST: a Novel Image Dataset for Benchmarking Machine Learning Algorithmscited by 4,283
  2. 2014Deep Learning Face Attributes in the Wildcited by 3,679
  3. 2017Compressed Sensing using Generative Modelscited by 337
  4. 2020Deep Learning Techniques for Inverse Problems in Imagingcited by 290
  5. 2014Phase Retrieval via Wirtinger Flow: Theory and Algorithmscited by 123
  6. 2018Deep Decoder: Concise Image Representations from Untrained Non-convolutional Networkscited by 120
  7. 2018Phase Retrieval Under a Generative Priorcited by 73
  8. 2013Phase Retrieval using Alternating Minimizationcited by 58
  9. 2018Solving Linear Inverse Problems Using GAN Priors: An Algorithm with Provable Guaranteescited by 58
  10. 2018Compressed Sensing with Deep Image Prior and Learned Regularizationcited by 47
  11. 2021Instance-Optimal Compressed Sensing via Posterior Samplingcited by 43
  12. 2018Optimality and Sub-optimality of PCA I: Spiked Random Matrix Modelscited by 33
21 more
  1. 2018Modeling Sparse Deviations for Compressed Sensing using Generative Modelscited by 33
  2. 2020Robust Compressed Sensing using Generative Modelscited by 33
  3. 2019Algorithmic Guarantees for Inverse Imaging with Untrained Network Priorscited by 32
  4. 2016Nonconvex phase synchronizationcited by 30
  5. 2014Statistical and computational trade-offs in estimation of sparse principal componentscited by 26
  6. 2019Information-Theoretic Lower Bounds for Compressive Sensing with Generative Modelscited by 25
  7. 2013Sparse PCA via Covariance Thresholdingcited by 21
  8. 2020Sample Complexity Bounds for 1-bit Compressive Sensing and Binary Stable Embeddings with Generative Priorscited by 21
  9. 2019The spiked matrix model with generative priorscited by 20
  10. 2019Exact asymptotics for phase retrieval and compressed sensing with random generative priorscited by 20
  11. 2019Alternating Phase Projected Gradient Descent with Generative Priors for Solving Compressive Phase Retrievalcited by 18
  12. 2016On the Estimation Performance and Convergence Rate of the Generalized Power Method for Phase Synchronizationcited by 17
  13. 2016The Projected Power Method: An Efficient Algorithm for Joint Alignment from Pairwise Differencescited by 15
  14. 2020The Generalized Lasso with Nonlinear Observations and Generative Priorscited by 15
  15. 2021Provable Compressed Sensing with Generative Priors via Langevin Dynamicscited by 14
  16. 2021Towards Sample-Optimal Compressive Phase Retrieval with Sparse and Generative Priorscited by 14
  17. 2013Sparse PCA through Low-rank Approximationscited by 11
  18. 2021Robust 1-bit Compressive Sensing with Partial Gaussian Circulant Matrices and Generative Priorscited by 7
  19. 2014A Convex Sparse PCA for Feature Analysiscited by 6
  20. 2017The Informativeness of K -Means for Learning Mixture Modelscited by 3
  21. 2020Non-Sparse PCA in High Dimensions via Cone Projected Power Iterationcited by 2

Built on by

Cited by year

Citing papers per year: 2022: 4; 2023: 4; 2024: 1; 2025: 3.

12 papers.

  1. 2023DiffFit: Unlocking Transferability of Large Diffusion Models via Simple Parameter-Efficient Fine-Tuningcited by 82
  2. 2022Theoretical Perspectives on Deep Learning Methods in Inverse Problemscited by 22
  3. 2022Non-Iterative Recovery from Nonlinear Observations using Generative Modelscited by 9
  4. 2022Misspecified Phase Retrieval with Generative Priorscited by 7
  5. 2025Integrating Intermediate Layer Optimization and Projected Gradient Descent for Solving Inverse Problems with Diffusion Modelscited by 7
  6. 2022Projected Gradient Descent Algorithms for Solving Nonlinear Inverse Problems with Generative Priorscited by 5
  7. 2023Solving Quadratic Systems with Full-Rank Matrices Using Sparse or Generative Priorscited by 4
  8. 2024Generalized Eigenvalue Problems with Generative Priorscited by 3
  9. 2025Learning Single Index Models with Diffusion Priorscited by 2
  10. 2023Do algorithms and barriers for sparse principal component analysis extend to other structured settings?cited by 1
  11. 2025Diffusion Model Based Signal Recovery Under 1-Bit Quantizationcited by 1
  12. 2023Using Intermediate Forward Iterates for Intermediate Generator Optimizationcited by 0