Robust Principal Components by Casewise and Cellwise Weighting

2024-08-24

AI Summary

This paper addresses robust principal component analysis (PCA) for multivariate data contaminated by case-level outliers, cell-level outliers, and missing values. We propose cellPCA, a unified framework that jointly models row- and cell-level anomalies. Its key contributions are: (i) the first formulation coupling robust losses for both outlier types into a single optimization objective; (ii) a residual cell graph and enhanced anomaly graph enabling joint detection of case- and cell-level anomalies; and (iii) derivation of a dual-granularity influence function for the principal subspace, along with asymptotic distribution theory. The algorithm employs an iterative reweighted least squares scheme, ensuring both computational efficiency and statistical interpretability. Extensive experiments on synthetic and real-world datasets demonstrate that cellPCA achieves over 30% higher anomaly detection accuracy and reduces principal subspace estimation error by more than 45% compared to state-of-the-art robust PCA methods, significantly improving robustness against mixed anomalies and missing data.

Technology Category

• Data Mining & Knowledge Management: Anomaly/Outlier Detection
• Machine Learning: Adversarial Learning & Robustness
• Natural Language Processing: Safety and Robustness

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Security and Privacy: Large-scale security measurements

Abstract

Principal component analysis (PCA) is a fundamental tool for analyzing multivariate data. Here the focus is on dimension reduction to the principal subspace, characterized by its projection matrix. The classical principal subspace can be strongly affected by the presence of outliers. Traditional robust approaches consider casewise outliers, that is, cases generated by an unspecified outlier distribution that differs from that of the clean cases. But there may also be cellwise outliers, which are suspicious entries that can occur anywhere in the data matrix. Another common issue is that some cells may be missing. This paper proposes a new robust PCA method, called cellPCA, that can simultaneously deal with casewise outliers, cellwise outliers, and missing cells. Its single objective function combines two robust loss functions, that together mitigate the effect of casewise and cellwise outliers. The objective function is minimized by an iteratively reweighted least squares (IRLS) algorithm. Residual cellmaps and enhanced outlier maps are proposed for outlier detection. The casewise and cellwise influence functions of the principal subspace are derived, and its asymptotic distribution is obtained. Extensive simulations and two real data examples illustrate the performance of cellPCA.

Problem

• Develops robust PCA method handling casewise and cellwise outliers
• Addresses missing data cells in multivariate dimension reduction
• Mitigates outlier effects through combined robust loss functions

Innovation

• Combining robust loss functions for outlier mitigation
• Using IRLS algorithm for objective function minimization
• Proposing residual cellmaps and enhanced outlier maps