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L1-Norm Principal Component Analysis Using Quaternion Rotations

Borowicz Adam

Annals of Computer Science and Information Systems

2023

Vol. 35

883--888

Principal component analysis (PCA) based on L1-norm has drawn growing interest in recent years. It is especially popular in the machine learning and pattern recognition communities for its robustness to outliers. Although optimal algorithms for L1-norm maximization exist, they have very high computational complexity and can be used for evaluation purposes only. In practice, only approximate techniques have been considered so far. Currently, the most popular method is the bit-flipping technique, where the L1-norm maximization is viewed as a combinatorial problem over the binary field. Recently, we proposed exhaustive, but faster algorithm based on two-dimensional Jacobi rotations that also offer high accuracy. In this paper, we develop a novel variant of this method that uses three-dimensional rotations and quaternion algebra. Our experiments show that the proposed approach offers higher accuracy than other approximate algorithms, but at the expense of the additional computational cost. However, for large datasets, the cost is still lower than that of the bit-flipping technique.