🤖 AI Summary
Selecting between PCA and SVD for dimensionality reduction of high-dimensional image data remains challenging due to ambiguity in their mathematical relationship and practical applicability. Method: Grounded in first principles of linear algebra, this work rigorously derives the mathematical foundations of both methods and systematically compares them across interpretability, numerical stability, and adaptability to non-square matrices. Contribution/Results: We establish, for the first time, a theory-driven, experiment-free selection criterion—precisely characterizing their intrinsic equivalence conditions (e.g., data centering) and operational boundaries (e.g., matrix aspect ratio, signal-to-noise ratio, floating-point precision constraints). By unifying classical matrix decomposition theory with modern numerical analysis, we identify the root causes of performance disparities, expose theoretical limitations, and prescribe a clear pathway for empirical validation. The results yield a universal, interpretable, and theoretically grounded algorithm selection framework for image dimensionality reduction.
📝 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. building on classical and recent numerical literature, 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.