# Statistical and computational trade-offs in estimation of sparse principal components

arXiv: https://arxiv.org/abs/1408.5369 · HTML: https://mlsanity.com/paper/1408.5369/

## Key facts

- Cited by 26 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 22 Aug 2014.

## Built on by (most cited first, 26 of 26)

- [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)
- [Statistical and Computational Trade-offs in Variational Inference: A Case Study in Inferential Model Selection](https://mlsanity.com/paper/2207.11208/) (2022)
- [Robust and Sparse Estimation of Linear Regression Coefficients with Heavy-tailed Noises and Covariates](https://mlsanity.com/paper/2206.07594/) (2022)
- [Higher degree sum-of-squares relaxations robust against oblivious outliers](https://mlsanity.com/paper/2211.07327/) (2022)
- [Computational Lower Bounds for Graphon Estimation via Low-degree Polynomials](https://mlsanity.com/paper/2308.15728/) (2023)
- [Outlier Robust and Sparse Estimation of Linear Regression Coefficients](https://mlsanity.com/paper/2208.11592/) (2022)
- [Random Subgraph Detection Using Queries](https://mlsanity.com/paper/2110.00744/) (2021)
- [Generalized Eigenvalue Problems with Generative Priors](https://mlsanity.com/paper/2411.01326/) (2024)
- [Kernel PCA with the Nystr"om method](https://mlsanity.com/paper/2109.05578/) (2021)
- [Inferring Hidden Structures in Random Graphs](https://mlsanity.com/paper/2110.01901/) (2021)
- [Matrix Reordering for Noisy Disordered Matrices: Optimality and Computationally Efficient Algorithms](https://mlsanity.com/paper/2201.06438/) (2022)
- [Planted Bipartite Graph Detection](https://mlsanity.com/paper/2302.03658/) (2023)
- [Sparse PCA Beyond Covariance Thresholding](https://mlsanity.com/paper/2302.10158/) (2023)
- [Robust Sparse Estimation for Gaussians with Optimal Error under Huber Contamination](https://mlsanity.com/paper/2403.10416/) (2024)
- [Do algorithms and barriers for sparse principal component analysis extend to other structured settings?](https://mlsanity.com/paper/2307.13535/) (2023)
- [Nonlinear Sparse Generalized Canonical Correlation Analysis for Multi-view High-dimensional Data](https://mlsanity.com/paper/2502.18756/) (2025)
- [On the Curse of Dimensionality in Private Sparse Covariance Estimation and PCA](https://mlsanity.com/paper/2606.21951/) (2026)
- [Logspace Reducibility From Secret Leakage Planted Clique](https://mlsanity.com/paper/2107.11886/) (2021)
- [Entrywise Recovery Guarantees for Sparse PCA via Sparsistent Algorithms](https://mlsanity.com/paper/2202.04061/) (2022)
- [Statistical and computational rates in high rank tensor estimation](https://mlsanity.com/paper/2304.04043/) (2023)
- [Sparse Linear Regression when Noises and Covariates are Heavy-Tailed and Contaminated by Outliers](https://mlsanity.com/paper/2408.01336/) (2024)
- [Enhancing Blind Source Separation with Dissociative Principal Component Analysis](https://mlsanity.com/paper/2411.12321/) (2024)
- [A Randomized Algorithm for Sparse PCA based on the Basic SDP Relaxation](https://mlsanity.com/paper/2507.09148/) (2025)
- [Combinatorial Sparse PCA Beyond the Spiked Identity Model](https://mlsanity.com/paper/2603.02607/) (2026)
- [End-to-End Efficient RL for Linear Bellman Complete MDPs with Deterministic Transitions](https://mlsanity.com/paper/2603.23461/) (2026)

## Abstract (arXiv)

In recent years, sparse principal component analysis has emerged as an extremely popular dimension reduction technique for high-dimensional data. The theoretical challenge, in the simplest case, is to estimate the leading eigenvector of a population covariance matrix under the assumption that this eigenvector is sparse. An impressive range of estimators have been proposed; some of these are fast to compute, while others are known to achieve the minimax optimal rate over certain Gaussian or sub-Gaussian classes. In this paper, we show that, under a widely-believed assumption from computational complexity theory, there is a fundamental trade-off between statistical and computational performance in this problem. More precisely, working with new, larger classes satisfying a restricted covariance concentration condition, we show that there is an effective sample size regime in which no randomised polynomial time algorithm can achieve the minimax optimal rate. We also study the theoretical performance of a (polynomial time) variant of the well-known semidefinite relaxation estimator, revealing a subtle interplay between statistical and computational efficiency.