mlsanityData to 7 Oct 2026

Inference for Heteroskedastic PCA with Missing Data

arXiv 2107.12365 ↗First posted 26 Jul 2021math.ST cs.IT cs.LGcited by 21

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Abstract (arXiv)

This paper studies how to construct confidence regions for principal component analysis (PCA) in high dimension, a problem that has been vastly under-explored. While computing measures of uncertainty for nonlinear/nonconvex estimators is in general difficult in high dimension, the challenge is further compounded by the prevalent presence of missing data and heteroskedastic noise. We propose a novel approach to performing valid inference on the principal subspace under a spiked covariance model with missing data, on the basis of an estimator called HeteroPCA (Zhang et al., 2022). We develop non-asymptotic distributional guarantees for HeteroPCA, and demonstrate how these can be invoked to compute both confidence regions for the principal subspace and entrywise confidence intervals for the spiked covariance matrix. Our inference procedures are fully data-driven and adaptive to heteroskedastic random noise, without requiring prior knowledge about the noise levels.

Key facts. Cited by 21 papers in the map (data through 7 Oct 2026). Builds on 37 of its 90 references that are in the map. First posted on arXiv 26 Jul 2021.

Builds on

37 of 90 references.

References · 37

  1. 2018Nonconvex Optimization Meets Low-Rank Matrix Factorization: An Overviewcited by 159
  2. 2020Spectral Methods for Data Science: A Statistical Perspectivecited by 111
  3. 2014Guaranteed Matrix Completion via Non-convex Factorizationcited by 81
  4. 2013Confidence Intervals and Hypothesis Testing for High-Dimensional Regressioncited by 78
  5. 2018Gradient Descent with Random Initialization: Fast Global Convergence for Nonconvex Phase Retrievalcited by 71
  6. 2015Fast low-rank estimation by projected gradient descent: General statistical and algorithmic guaranteescited by 68
  7. 2017Implicit Regularization in Nonconvex Statistical Estimation: Gradient Descent Converges Linearly for Phase Retrieval, Matrix Completion, and Blind Deconvolutioncited by 63
  8. 2019Noisy Matrix Completion: Understanding Statistical Guarantees for Convex Relaxation via Nonconvex Optimizationcited by 58
  9. 2017Spectral Method and Regularized MLE Are Both Optimal for Top-K Rankingcited by 46
  10. 2016Rate-Optimal Perturbation Bounds for Singular Subspaces with Applications to High-Dimensional Statisticscited by 45
  11. 2020The Lasso with general Gaussian designs with applications to hypothesis testingcited by 40
  12. 2019Inference and Uncertainty Quantification for Noisy Matrix Completioncited by 39
25 more
  1. 2016Convergence Analysis for Rectangular Matrix Completion Using Burer-Monteiro Factorization and Gradient Descentcited by 37
  2. 2019Nonconvex Low-Rank Tensor Completion from Noisy Datacited by 35
  3. 2018Heteroskedastic PCA: Algorithm, Optimality, and Applicationscited by 33
  4. 2016Spectral algorithms for tensor completioncited by 27
  5. 2014A General Theory of Hypothesis Tests and Confidence Regions for Sparse High Dimensional Modelscited by 26
  6. 2020Bridging Convex and Nonconvex Optimization in Robust PCA: Noise, Outliers, and Missing Datacited by 23
  7. 2018Streaming PCA and Subspace Tracking: The Missing Data Casecited by 18
  8. 2019Statistical Inferences of Linear Forms for Noisy Matrix Completioncited by 18
  9. 2020Near-Optimal Performance Bounds for Orthogonal and Permutation Group Synchronization via Spectral Methodscited by 18
  10. 2017Unperturbed: spectral analysis beyond Davis-Kahancited by 16
  11. 2019High-dimensional principal component analysis with heterogeneous missingnesscited by 16
  12. 2018Robust high dimensional factor models with applications to statistical machine learningcited by 13
  13. 2020Uncertainty quantification for nonconvex tensor completion: Confidence intervals, heteroscedasticity and optimalitycited by 13
  14. 2013Asymptotic normality and optimalities in estimation of large Gaussian graphical modelscited by 12
  15. 2021Entrywise Estimation of Singular Vectors of Low-Rank Matrices with Heteroskedasticity and Dependencecited by 12
  16. 2018Asymmetry Helps: Eigenvalue and Eigenvector Analyses of Asymmetrically Perturbed Low-Rank Matricescited by 10
  17. 2015Spectral Thresholds in the Bipartite Stochastic Block Modelcited by 9
  18. 2020Tackling small eigen-gaps: Fine-grained eigenvector estimation and inference under heteroscedastic noisecited by 9
  19. 2021Inference for Low-Rank Modelscited by 7
  20. 2018Adapting to Unknown Noise Distribution in Matrix Denoisingcited by 5
  21. 2015Asymptotic Theory for Estimating the Singular Vectors and Values of a Partially-observed Low Rank Matrix with Noisecited by 3
  22. 2019Subspace Estimation from Unbalanced and Incomplete Data Matrices: ℓ2,∞ Statistical Guaranteescited by 2
  23. 2020Recent Developments on Factor Models and its Applications in Econometric Learningcited by 2
  24. 2019Nonconvex Rectangular Matrix Completion via Gradient Descent without ℓ2,∞ Regularizationcited by 1
  25. 2021Minimax Estimation of Linear Functions of Eigenvectors in the Face of Small Eigen-Gapscited by 1

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Cited by year (2026 to 7 Oct)

Citing papers per year: 2022: 3; 2023: 5; 2024: 7; 2025: 2; 2026: 4.

21 papers.

Sort:

  1. 2023Understanding Multimodal Contrastive Learning and Incorporating Unpaired Datacited by 43
  2. 2023Approximate message passing from random initialization with applications to mathbbZ2 synchronizationcited by 9
  3. 2022Estimating Higher-Order Mixed Memberships via the ℓ2,∞ Tensor Perturbation Boundcited by 8
  4. 2023Deflated HeteroPCA: Overcoming the curse of ill-conditioning in heteroskedastic PCAcited by 7
  5. 2022Robust Matrix Completion with Heavy-tailed Noisecited by 4
  6. 2024High-dimensional logistic regression with missing data: Imputation, regularization, and universalitycited by 4
  7. 2023Heteroskedastic Tensor Clusteringcited by 3
  8. 2024A non-asymptotic distributional theory of approximate message passing for sparse and robust regressioncited by 3
  9. 2024Online Policy Learning and Inference by Matrix Completioncited by 3
  10. 2024LEARNER: A Transfer Learning Method for Low-Rank Matrix Estimationcited by 3
  11. 2023Inferences on Mixing Probabilities and Ranking in Mixed-Membership Modelscited by 1
  12. 2022Entrywise Recovery Guarantees for Sparse PCA via Sparsistent Algorithmscited by 0
9 more
  1. 2024On Minimum Trace Factor Analysis -- An Old Song Sung to a New Tunecited by 0
  2. 2024Bias-Corrected Joint Spectral Embedding for Multilayer Networks with Invariant Subspace: Entrywise Eigenvector Perturbation and Inferencecited by 0
  3. 2024Generalized Grade-of-Membership Estimation for High-dimensional Locally Dependent Datacited by 0
  4. 2025An Interdisciplinary and Cross-Task Review on Missing Data Imputationcited by 0
  5. 2025Inference for Similarity and Alignability between Noisy High-Dimensional Datasetscited by 0
  6. 2026Adjacency Spectral Embeddings of Correlation Networkscited by 0
  7. 2026High-Dimensional Gaussian Mean Estimation under Realizable Contaminationcited by 0
  8. 2026Unveiling Invariant and Transferable Latent Factors Across Heterogeneous Environments via ATLAScited by 0
  9. 2026Toward a Unified Statistical Theory of Unsupervised Pretraining and Supervised Neural Knowledge Graph Learningcited by 0