# Inference for Heteroskedastic PCA with Missing Data

arXiv: https://arxiv.org/abs/2107.12365 · HTML: https://mlsanity.com/paper/2107.12365/

## Key facts

- Cited by 21 papers in the map (data through 14 Oct 2026).
- Builds on 37 of its 90 references that are in the map.
- First posted on arXiv 26 Jul 2021.

## Builds on (most cited first, 30 of 37)

- [Nonconvex Optimization Meets Low-Rank Matrix Factorization: An Overview](https://mlsanity.com/paper/1809.09573/) (2018)
- [Spectral Methods for Data Science: A Statistical Perspective](https://mlsanity.com/paper/2012.08496/) (2020)
- [Guaranteed Matrix Completion via Non-convex Factorization](https://mlsanity.com/paper/1411.8003/) (2014)
- [Confidence Intervals and Hypothesis Testing for High-Dimensional Regression](https://mlsanity.com/paper/1306.3171/) (2013)
- [Gradient Descent with Random Initialization: Fast Global Convergence for Nonconvex Phase Retrieval](https://mlsanity.com/paper/1803.07726/) (2018)
- [Fast low-rank estimation by projected gradient descent: General statistical and algorithmic guarantees](https://mlsanity.com/paper/1509.03025/) (2015)
- [Implicit Regularization in Nonconvex Statistical Estimation: Gradient Descent Converges Linearly for Phase Retrieval, Matrix Completion, and Blind Deconvolution](https://mlsanity.com/paper/1711.10467/) (2017)
- [Noisy Matrix Completion: Understanding Statistical Guarantees for Convex Relaxation via Nonconvex Optimization](https://mlsanity.com/paper/1902.07698/) (2019)
- [Spectral Method and Regularized MLE Are Both Optimal for Top-K Ranking](https://mlsanity.com/paper/1707.09971/) (2017)
- [Rate-Optimal Perturbation Bounds for Singular Subspaces with Applications to High-Dimensional Statistics](https://mlsanity.com/paper/1605.00353/) (2016)
- [The Lasso with general Gaussian designs with applications to hypothesis testing](https://mlsanity.com/paper/2007.13716/) (2020)
- [Convergence Analysis for Rectangular Matrix Completion Using Burer-Monteiro Factorization and Gradient Descent](https://mlsanity.com/paper/1605.07051/) (2016)
- [Inference and Uncertainty Quantification for Noisy Matrix Completion](https://mlsanity.com/paper/1906.04159/) (2019)
- [Nonconvex Low-Rank Tensor Completion from Noisy Data](https://mlsanity.com/paper/1911.04436/) (2019)
- [Heteroskedastic PCA: Algorithm, Optimality, and Applications](https://mlsanity.com/paper/1810.08316/) (2018)
- [Spectral algorithms for tensor completion](https://mlsanity.com/paper/1612.07866/) (2016)
- [A General Theory of Hypothesis Tests and Confidence Regions for Sparse High Dimensional Models](https://mlsanity.com/paper/1412.8765/) (2014)
- [Bridging Convex and Nonconvex Optimization in Robust PCA: Noise, Outliers, and Missing Data](https://mlsanity.com/paper/2001.05484/) (2020)
- [Streaming PCA and Subspace Tracking: The Missing Data Case](https://mlsanity.com/paper/1806.04609/) (2018)
- [Statistical Inferences of Linear Forms for Noisy Matrix Completion](https://mlsanity.com/paper/1909.00116/) (2019)
- [Near-Optimal Performance Bounds for Orthogonal and Permutation Group Synchronization via Spectral Methods](https://mlsanity.com/paper/2008.05341/) (2020)
- [Unperturbed: spectral analysis beyond Davis-Kahan](https://mlsanity.com/paper/1706.06516/) (2017)
- [High-dimensional principal component analysis with heterogeneous missingness](https://mlsanity.com/paper/1906.12125/) (2019)
- [Robust high dimensional factor models with applications to statistical machine learning](https://mlsanity.com/paper/1808.03889/) (2018)
- [Uncertainty quantification for nonconvex tensor completion: Confidence intervals, heteroscedasticity and optimality](https://mlsanity.com/paper/2006.08580/) (2020)
- [Asymptotic normality and optimalities in estimation of large Gaussian graphical models](https://mlsanity.com/paper/1309.6024/) (2013)
- [Entrywise Estimation of Singular Vectors of Low-Rank Matrices with Heteroskedasticity and Dependence](https://mlsanity.com/paper/2105.13346/) (2021)
- [Asymmetry Helps: Eigenvalue and Eigenvector Analyses of Asymmetrically Perturbed Low-Rank Matrices](https://mlsanity.com/paper/1811.12804/) (2018)
- [Spectral Thresholds in the Bipartite Stochastic Block Model](https://mlsanity.com/paper/1506.06737/) (2015)
- [Tackling small eigen-gaps: Fine-grained eigenvector estimation and inference under heteroscedastic noise](https://mlsanity.com/paper/2001.04620/) (2020)

## Built on by (most cited first, 21 of 21)

- [Understanding Multimodal Contrastive Learning and Incorporating Unpaired Data](https://mlsanity.com/paper/2302.06232/) (2023)
- [Approximate message passing from random initialization with applications to mathbbZ2 synchronization](https://mlsanity.com/paper/2302.03682/) (2023)
- [Estimating Higher-Order Mixed Memberships via the ℓ2,∞ Tensor Perturbation Bound](https://mlsanity.com/paper/2212.08642/) (2022)
- [Deflated HeteroPCA: Overcoming the curse of ill-conditioning in heteroskedastic PCA](https://mlsanity.com/paper/2303.06198/) (2023)
- [High-dimensional logistic regression with missing data: Imputation, regularization, and universality](https://mlsanity.com/paper/2410.01093/) (2024)
- [Robust Matrix Completion with Heavy-tailed Noise](https://mlsanity.com/paper/2206.04276/) (2022)
- [A non-asymptotic distributional theory of approximate message passing for sparse and robust regression](https://mlsanity.com/paper/2401.03923/) (2024)
- [Heteroskedastic Tensor Clustering](https://mlsanity.com/paper/2311.02306/) (2023)
- [Online Policy Learning and Inference by Matrix Completion](https://mlsanity.com/paper/2404.17398/) (2024)
- [LEARNER: A Transfer Learning Method for Low-Rank Matrix Estimation](https://mlsanity.com/paper/2412.20605/) (2024)
- [Inferences on Mixing Probabilities and Ranking in Mixed-Membership Models](https://mlsanity.com/paper/2308.14988/) (2023)
- [Inference for Similarity and Alignability between Noisy High-Dimensional Datasets](https://mlsanity.com/paper/2511.21074/) (2025)
- [High-Dimensional Gaussian Mean Estimation under Realizable Contamination](https://mlsanity.com/paper/2603.16798/) (2026)
- [Entrywise Recovery Guarantees for Sparse PCA via Sparsistent Algorithms](https://mlsanity.com/paper/2202.04061/) (2022)
- [On Minimum Trace Factor Analysis -- An Old Song Sung to a New Tune](https://mlsanity.com/paper/2402.02459/) (2024)
- [Bias-Corrected Joint Spectral Embedding for Multilayer Networks with Invariant Subspace: Entrywise Eigenvector Perturbation and Inference](https://mlsanity.com/paper/2406.07849/) (2024)
- [Generalized Grade-of-Membership Estimation for High-dimensional Locally Dependent Data](https://mlsanity.com/paper/2412.19796/) (2024)
- [An Interdisciplinary and Cross-Task Review on Missing Data Imputation](https://mlsanity.com/paper/2511.01196/) (2025)
- [Adjacency Spectral Embeddings of Correlation Networks](https://mlsanity.com/paper/2602.21055/) (2026)
- [Unveiling Invariant and Transferable Latent Factors Across Heterogeneous Environments via ATLAS](https://mlsanity.com/paper/2607.18209/) (2026)
- [Toward a Unified Statistical Theory of Unsupervised Pretraining and Supervised Neural Knowledge Graph Learning](https://mlsanity.com/paper/2607.26346/) (2026)

## 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.