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This work aims to explore the significance of Singular Value Decomposition and spectral methods in modern AI systems.

June 5, 2026Open Access

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

Key Points

This work aims to explore the significance of Singular Value Decomposition and spectral methods in modern AI systems.
Discussion of SVD foundations in vector spaces and orthogonality
Analysis of applications including principal component analysis and recommendation systems
Examination of computational challenges in large language models
SVD and spectral methods effectively reduce dimensionality and enhance feature extraction in AI applications
Low-rank spectral structures improve the performance of neural architectures
Computational challenges in large language models necessitate continued research in low-rank methods

Abstract

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.

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