# Entrywise Recovery Guarantees for Sparse PCA via Sparsistent Algorithms

arXiv: https://arxiv.org/abs/2202.04061 · HTML: https://mlsanity.com/paper/2202.04061/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 21 of its 71 references that are in the map.
- First posted on arXiv 8 Feb 2022.

## Builds on (most cited first, 21 of 21)

- [Spectral Methods for Data Science: A Statistical Perspective](https://mlsanity.com/paper/2012.08496/) (2020)
- [Noisy Matrix Completion: Understanding Statistical Guarantees for Convex Relaxation via Nonconvex Optimization](https://mlsanity.com/paper/1902.07698/) (2019)
- [Rate-Optimal Perturbation Bounds for Singular Subspaces with Applications to High-Dimensional Statistics](https://mlsanity.com/paper/1605.00353/) (2016)
- [Inference and Uncertainty Quantification for Noisy Matrix Completion](https://mlsanity.com/paper/1906.04159/) (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)
- [Estimating Mixed Memberships with Sharp Eigenvector Deviations](https://mlsanity.com/paper/1709.00407/) (2017)
- [Inference for Heteroskedastic PCA with Missing Data](https://mlsanity.com/paper/2107.12365/) (2021)
- [Statistical Inferences of Linear Forms for Noisy Matrix Completion](https://mlsanity.com/paper/1909.00116/) (2019)
- [Vintage Factor Analysis with Varimax Performs Statistical Inference](https://mlsanity.com/paper/2004.05387/) (2020)
- [Dimensionality reduction, regularization, and generalization in overparameterized regressions](https://mlsanity.com/paper/2011.11477/) (2020)
- [Entrywise Estimation of Singular Vectors of Low-Rank Matrices with Heteroskedasticity and Dependence](https://mlsanity.com/paper/2105.13346/) (2021)
- [Do semidefinite relaxations solve sparse PCA up to the information limit?](https://mlsanity.com/paper/1306.3690/) (2013)
- [Asymptotically efficient estimators for stochastic blockmodels: the naive MLE, the rank-constrained MLE, and the spectral](https://mlsanity.com/paper/1710.10936/) (2017)
- [Sparsistency and agnostic inference in sparse PCA](https://mlsanity.com/paper/1401.6978/) (2014)
- [The eigenvalues of stochastic blockmodel graphs](https://mlsanity.com/paper/1803.11551/) (2018)
- [Sparse spectral estimation with missing and corrupted measurements](https://mlsanity.com/paper/1811.10443/) (2018)
- [An ℓp theory of PCA and spectral clustering](https://mlsanity.com/paper/2006.14062/) (2020)
- [Sparse Nonlinear Regression: Parameter Estimation and Asymptotic Inference](https://mlsanity.com/paper/1511.04514/) (2015)
- [Bayesian Estimation of Sparse Spiked Covariance Matrices in High Dimensions](https://mlsanity.com/paper/1808.07433/) (2018)
- [Entrywise convergence of iterative methods for eigenproblems](https://mlsanity.com/paper/2002.08491/) (2020)

## Abstract (arXiv)

Sparse Principal Component Analysis (PCA) is a prevalent tool across a plethora of subfields of applied statistics. While several results have characterized the recovery error of the principal eigenvectors, these are typically in spectral or Frobenius norms. In this paper, we provide entrywise ℓ2,∞ bounds for Sparse PCA under a general high-dimensional subgaussian design. In particular, our results hold for any algorithm that selects the correct support with high probability, those that are sparsistent. Our bound improves upon known results by providing a finer characterization of the estimation error, and our proof uses techniques recently developed for entrywise subspace perturbation theory.