Outlier-resistant data processing with L1-norm principal component analysis

Abstract

Principal Component Analysis (PCA) has been a cornerstone of data analysis for more than a century, with important applications across most fields of science and engineering. However, despite its many strengths, PCA is known to have amajor drawback: it is very sensitive to the presence of outliers among the processed data. To counteract the impact of outliers in data analysis, researchers have been long working on robustmodifications of PCA. One of the most successful (and promising) PCA alternatives is L1-PCA. L1-PCA relies on the L1-norm of the processed data and, thus, tames any outliers that may exist in the dataset. Experimental studies in various applications have shown that L1-PCA (i) attains similar performance to PCA when the processed data are outlier-free and (ii) maintains sturdy resistance against outliers when the processed data are corrupted. Thus, L1-PCA is expected to play a significant role in the big-data era, when large datasets are often outlier corrupted. In this chapter, we present the theoretical foundations of L1-PCA, optimal and state-of- the-art approximate algorithms for its implementation, and some numerical studies that demonstrate its favorable performance.