Singular Value Decomposition and Spectral Methods for Low-Rank Learning in Modern Artificial Intelligence

Efficient mathematical modelling and optimisation are essential for modern computational learning systems, as Artificial Intelligence and Machine Learning systems are increasingly dependent on high-dimensional data representations and large-scale neural architectures. Singular Value Decomposition (SVD) and spectral methods are mathematical tools in AI, and have become powerful techniques for dimensionality reduction, latent feature extraction, and low-rank approximation. In this paper, we discuss the mathematical basis of SVD from the perspective of vector spaces, orthogonality, matrix transformations, eigenvalue analysis and spectral decomposition theory and analyse their importance in modern Artificial Intelligence systems. We study the role of low-rank spectral structures in applications such as principal component analysis, recommendation systems, image compression, natural language processing and neural representation learning. Special attention is devoted to the computational challenges posed by Large Language Models. Keywords—Singular Value Decomposition; Spectral Methods; Low-Rank Learning; Artificial Intelligence; Large Language Models.