# A Linearly Convergent Algorithm for Rotationally Invariant ℓ1-Norm Principal Component Analysis

arXiv: https://arxiv.org/abs/2210.05066 · HTML: https://mlsanity.com/paper/2210.05066/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 11 Oct 2022.

## Built on by (most cited first, 2 of 2)

- [On Finite-Step Convergence of the Non-Greedy Algorithm and Proximal Alternating Minimization Method with Extrapolation for L1-Norm PCA](https://mlsanity.com/paper/2302.07712/) (2023)
- [ℓ1-norm rank-one symmetric matrix factorization has no spurious second-order stationary points](https://mlsanity.com/paper/2410.05025/) (2024)

## Abstract (arXiv)

To do dimensionality reduction on the datasets with outliers, the ℓ1-norm principal component analysis (L1-PCA) as a typical robust alternative of the conventional PCA has enjoyed great popularity over the past years. In this work, we consider a rotationally invariant L1-PCA, which is hardly studied in the literature. To tackle it, we propose a proximal alternating linearized minimization method with a nonlinear extrapolation for solving its two-block reformulation. Moreover, we show that the proposed method converges at least linearly to a limiting critical point of the reformulated problem. Such a point is proved to be a critical point of the original problem under a condition imposed on the step size. Finally, we conduct numerical experiments on both synthetic and real datasets to support our theoretical developments and demonstrate the efficacy of our approach.