# Computational Lower Bounds for Sparse PCA

arXiv: https://arxiv.org/abs/1304.0828 · HTML: https://mlsanity.com/paper/1304.0828/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 3 Apr 2013.

## Built on by (most cited first, 11 of 11)

- [Lattice-Based Methods Surpass Sum-of-Squares in Clustering](https://mlsanity.com/paper/2112.03898/) (2021)
- [Statistical and Computational Phase Transitions in Group Testing](https://mlsanity.com/paper/2206.07640/) (2022)
- [Higher degree sum-of-squares relaxations robust against oblivious outliers](https://mlsanity.com/paper/2211.07327/) (2022)
- [Sparse GCA and Thresholded Gradient Descent](https://mlsanity.com/paper/2107.00371/) (2021)
- [Cryptographic Hardness of Score Estimation](https://mlsanity.com/paper/2404.03272/) (2024)
- [Sparse PCA Beyond Covariance Thresholding](https://mlsanity.com/paper/2302.10158/) (2023)
- [Low-Degree Method Fails to Predict Robust Subspace Recovery](https://mlsanity.com/paper/2603.02594/) (2026)
- [Sparse PCA with Oracle Property](https://mlsanity.com/paper/2312.16793/) (2023)
- [A Randomized Algorithm for Sparse PCA based on the Basic SDP Relaxation](https://mlsanity.com/paper/2507.09148/) (2025)
- [Backdoor Channels Hidden in Latent Space: Extending Cryptographic Undetectability to Modern Neural Networks](https://mlsanity.com/paper/2605.13214/) (2026)
- [Implementing a White-Box Undetectable Backdoor for Random Fourier Features](https://mlsanity.com/paper/2609.16403/) (2026)

## Abstract (arXiv)

In the context of sparse principal component detection, we bring evidence towards the existence of a statistical price to pay for computational efficiency. We measure the performance of a test by the smallest signal strength that it can detect and we propose a computationally efficient method based on semidefinite programming. We also prove that the statistical performance of this test cannot be strictly improved by any computationally efficient method. Our results can be viewed as complexity theoretic lower bounds conditionally on the assumptions that some instances of the planted clique problem cannot be solved in randomized polynomial time.