A Comparative Analysis of Principal Component Analysis (PCA) and Singular Value Decomposition (SVD) as Dimensionality Reduction Techniques

TL;DR

This paper analytically compares PCA and SVD, two linear techniques for dimensionality reduction, focusing on interpretability, stability, and applicability, providing guidelines without empirical testing.

Contribution

It offers a theoretical comparison of PCA and SVD, deriving algorithms from first principles and establishing criteria for choosing between them based on classical and recent literature.

Findings

PCA and SVD have different interpretability and stability profiles.

Guidelines for selecting PCA or SVD based on matrix shape and data characteristics.

No empirical benchmarking; focus on theoretical analysis.

Abstract

High-dimensional image data often require dimensionality reduction before further analysis. This paper provides a purely analytical comparison of two linear techniques-Principal Component Analysis (PCA) and Singular Value Decomposition (SVD). After the derivation of each algorithm from first principles, we assess their interpretability, numerical stability, and suitability for differing matrix shapes. We synthesize rule-of-thumb guidelines for choosing one out of the two algorithms without empirical benchmarking, building on classical and recent numerical literature. Limitations and directions for future experimental work are outlined at the end.