Principal Component Analysis Based on T$\ell_1$-norm Maximization

TL;DR

This paper introduces a PCA method based on T$ ilde{ ext{l}}_1$-norm, which offers improved robustness to outliers and noise, outperforming existing $ ext{l}_p$-norm and $ ext{l}_1$-norm PCA methods in experiments.

Contribution

The paper proposes a novel PCA approach using T$ ilde{ ext{l}}_1$-norm, demonstrating superior robustness and performance over traditional and $ ext{l}_p$-norm PCA methods.

Findings

T$ ilde{ ext{l}}_1$-norm PCA outperforms PCA-$ ext{l}_p$ and PCA-$ ext{l}_1$ in robustness.

Numerical experiments confirm the superior performance of the proposed method.

The method effectively suppresses outliers and noise.

Abstract

Classical principal component analysis (PCA) may suffer from the sensitivity to outliers and noise. Therefore PCA based on $\ell_1$-norm and $\ell_p$-norm ($0 < p < 1$) have been studied. Among them, the ones based on $\ell_p$-norm seem to be most interesting from the robustness point of view. However, their numerical performance is not satisfactory. Note that, although T$\ell_1$-norm is similar to $\ell_p$-norm ($0 < p < 1$) in some sense, it has the stronger suppression effect to outliers and better continuity. So PCA based on T$\ell_1$-norm is proposed in this paper. Our numerical experiments have shown that its performance is superior than PCA-$\ell_p$ and $\ell_p$SPCA as well as PCA, PCA-$\ell_1$ obviously.