Robust Principal Component Analysis Using a Novel Kernel Related with the L1-Norm

TL;DR

This paper introduces a robust PCA method based on a novel kernel related to the L1-norm, leveraging energy-efficient, multiplication-free vector dot products that enhance robustness to impulsive noise.

Contribution

The paper proposes a new energy-efficient, multiplication-free kernel for PCA that induces the L1-norm, enabling robust analysis with improved noise resilience and computational efficiency.

Findings

Achieves higher peak signal-to-noise ratios in image reconstruction.

Provides a mathematically proven positive semi-definite covariance matrix.

Demonstrates robustness to impulsive noise in practical applications.

Abstract

We consider a family of vector dot products that can be implemented using sign changes and addition operations only. The dot products are energy-efficient as they avoid the multiplication operation entirely. Moreover, the dot products induce the $\ell_1$-norm, thus providing robustness to impulsive noise. First, we analytically prove that the dot products yield symmetric, positive semi-definite generalized covariance matrices, thus enabling principal component analysis (PCA). Moreover, the generalized covariance matrices can be constructed in an Energy Efficient (EEF) manner due to the multiplication-free property of the underlying vector products. We present image reconstruction examples in which our EEF PCA method result in the highest peak signal-to-noise ratios compared to the ordinary $\ell_2$-PCA and the recursive $\ell_1$-PCA.