# Finite Sample Guarantees for PCA in Non-Isotropic and Data-Dependent Noise

arXiv: https://arxiv.org/abs/1709.06255 · HTML: https://mlsanity.com/paper/1709.06255/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 19 Sep 2017.

## Built on by (most cited first, 3 of 3)

- [Subspace Recovery from Heterogeneous Data with Non-isotropic Noise](https://mlsanity.com/paper/2210.13497/) (2022)
- [Capturing the Denoising Effect of PCA via Compression Ratio](https://mlsanity.com/paper/2204.10888/) (2022)
- [SuperPCA: subspace analysis and an efficient algorithm for high-dimensional PCA](https://mlsanity.com/paper/2609.26406/) (2026)

## Abstract (arXiv)

This work obtains novel finite sample guarantees for Principal Component Analysis (PCA). These hold even when the corrupting noise is non-isotropic, and a part (or all of it) is data-dependent. Because of the latter, in general, the noise and the true data are correlated. The results in this work are a significant improvement over those given in our earlier work where this "correlated-PCA" problem was first studied. In fact, in certain regimes, our results imply that the sample complexity required to achieve subspace recovery error that is a constant fraction of the noise level is near-optimal. Useful corollaries of our result include guarantees for PCA in sparse data-dependent noise and for PCA with missing data. An important application of the former is in proving correctness of the subspace update step of a popular online algorithm for dynamic robust PCA.