Extended Principal Component Analysis

TL;DR

This paper introduces a three-term PCA extension that enhances accuracy and reduces computational load, applicable even to singular data, supported by theoretical proof and real-world simulations.

Contribution

It proposes a novel three-term PCA transform with a special structure, improving upon existing PCA techniques in accuracy and efficiency.

Findings

The three-term PCA always exists and is applicable to singular data.

The method improves reconstruction accuracy compared to traditional PCA.

Simulations demonstrate effectiveness on real-world datasets.

Abstract

Principal Component Analysis (PCA) is a transform for finding the principal components (PCs) that represent features of random data. PCA also provides a reconstruction of the PCs to the original data. We consider an extension of PCA which allows us to improve the associated accuracy and diminish the numerical load, in comparison with known techniques. This is achieved due to the special structure of the proposed transform which contains two matrices $T_0$ and $T_1$, and a special transformation $\mathcal{f}$ of the so called auxiliary random vector $\mathbf w$. For this reason, we call it the three-term PCA. In particular, we show that the three-term PCA always exists, i.e. is applicable to the case of singular data. Both rigorous theoretical justification of the three-term PCA and simulations with real-world data are provided.