Robust Orthogonal Complement Principal Component Analysis

TL;DR

This paper introduces ROC-PCA, a robust method for principal component analysis that effectively handles complex outliers through a novel mathematical framework, combining sparsity and low-rank regularization, with proven accuracy and computational efficiency.

Contribution

The paper proposes ROC-PCA, a new robust PCA method that addresses transformed outliers using a combined regularization approach and develops efficient algorithms for high-dimensional data.

Findings

ROC-PCA outperforms existing methods in synthetic experiments.

It demonstrates high breakdown performance and accuracy.

The method is effective on real-world datasets.

Abstract

Recently, the robustification of principal component analysis has attracted lots of attention from statisticians, engineers and computer scientists. In this work we study the type of outliers that are not necessarily apparent in the original observation space but can seriously affect the principal subspace estimation. Based on a mathematical formulation of such transformed outliers, a novel robust orthogonal complement principal component analysis (ROC-PCA) is proposed. The framework combines the popular sparsity-enforcing and low rank regularization techniques to deal with row-wise outliers as well as element-wise outliers. A non-asymptotic oracle inequality guarantees the accuracy and high breakdown performance of ROC-PCA in finite samples. To tackle the computational challenges, an efficient algorithm is developed on the basis of Stiefel manifold optimization and iterative thresholding. Furthermore, a batch variant is proposed to significantly reduce the cost in ultra high dimensions. The paper also points out a pitfall of a common practice of SVD reduction in robust PCA. Experiments show the effectiveness and efficiency of ROC-PCA in both synthetic and real data.