Robust Principal Component Analysis Using a Novel Kernel Related with the L1-Norm
This work addresses robust PCA for image processing, offering an energy-efficient solution, but it is incremental as it builds on existing L1-norm methods.
The paper tackles robust principal component analysis (PCA) by introducing a family of multiplication-free vector dot products that induce the L1-norm, providing robustness to impulsive noise. The method achieves the highest peak signal-to-noise ratios in image reconstruction compared to ordinary L2-PCA and recursive L1-PCA.
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.