A Pure L1-norm Principal Component Analysis.

Area of Science:

• Statistics
• Machine Learning
• Data Analysis

Background:

• Principal Component Analysis (PCA) traditionally uses the L2 norm, which is sensitive to outliers.
• L1-norm PCA offers robustness against outliers and is suitable for non-Gaussian noise models.
• Existing L1-norm PCA methods often fail to provide globally optimal solutions efficiently.

Purpose of the Study:

• To propose a novel L1-norm PCA procedure that achieves globally optimal solutions in polynomial time.
• To introduce L1-PCA*, an efficient method based on the L1-norm best-fit hyperplane problem.
• To demonstrate the effectiveness of L1-PCA* in handling outlier-contaminated data.

Main Methods:

• Recasting L1-norm PCA as an optimization problem.
• Efficiently calculating the optimal solution for the L1-norm best-fit hyperplane.
• Developing the L1-PCA* algorithm for fitting data to successively smaller dimensional subspaces.

Main Results:

• L1-PCA* provides globally optimal solutions for L1-norm PCA.
• The procedure is implemented and tested on various datasets.
• L1-PCA* demonstrates superior performance in the presence of unbalanced outlier contamination.

Conclusions:

• L1-PCA* is an efficient and globally optimal L1-norm PCA method.
• The proposed method is particularly advantageous for datasets with outliers.
• L1-PCA* offers a robust alternative to traditional L2-based PCA.