# Square Root Principal Component Pursuit: Tuning-Free Noisy Robust Matrix Recovery

arXiv: https://arxiv.org/abs/2106.09211 · HTML: https://mlsanity.com/paper/2106.09211/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 7 of its 34 references that are in the map.
- First posted on arXiv 17 Jun 2021.

## Builds on (most cited first, 7 of 7)

- [No Spurious Local Minima in Nonconvex Low Rank Problems: A Unified Geometric Analysis](https://mlsanity.com/paper/1704.00708/) (2017)
- [Fast low-rank estimation by projected gradient descent: General statistical and algorithmic guarantees](https://mlsanity.com/paper/1509.03025/) (2015)
- [Non-convex Robust PCA](https://mlsanity.com/paper/1410.7660/) (2014)
- [Robust Recovery via Implicit Bias of Discrepant Learning Rates for Double Over-parameterization](https://mlsanity.com/paper/2006.08857/) (2020)
- [Bridging Convex and Nonconvex Optimization in Robust PCA: Noise, Outliers, and Missing Data](https://mlsanity.com/paper/2001.05484/) (2020)
- [Matrix Completion with Noisy Entries and Outliers](https://mlsanity.com/paper/1503.00214/) (2015)
- [The flare Package for High Dimensional Linear Regression and Precision Matrix Estimation in R](https://mlsanity.com/paper/2006.15419/) (2020)

## Built on by (most cited first, 1 of 1)

- [Alternating minimization for square root principal component pursuit](https://mlsanity.com/paper/2501.00471/) (2024)

## Abstract (arXiv)

We propose a new framework -- Square Root Principal Component Pursuit -- for low-rank matrix recovery from observations corrupted with noise and outliers. Inspired by the square root Lasso, this new formulation does not require prior knowledge of the noise level. We show that a single, universal choice of the regularization parameter suffices to achieve reconstruction error proportional to the (a priori unknown) noise level. In comparison, previous formulations such as stable PCP rely on noise-dependent parameters to achieve similar performance, and are therefore challenging to deploy in applications where the noise level is unknown. We validate the effectiveness of our new method through experiments on simulated and real datasets. Our simulations corroborate the claim that a universal choice of the regularization parameter yields near optimal performance across a range of noise levels, indicating that the proposed method outperforms the (somewhat loose) bound proved here.