Mathematical Foundations of Machine Learning: A Linear Algebra, Probability and Optimisation Perspective
Year: 2026
Type: book-chapter
Abstract: Machine learning has emerged as one of the most influential branches of modern engineering, driving advances in automation, prediction, and intelligent decision-making across domains such as manufacturing, energy, communication, and civil infrastructure. While considerable attention is given to algorithms, software libraries, and computational tools, the underlying mathematical structure that makes learning possible is often treated superficially in applied engineering literature. This chapter presents an integrated mathematical account of machine learning, organized around three foundational pillars: linear algebra, probability and statistics, and optimisation theory. Vector spaces, matrix transformations, and eigenvalue decomposition are examined as the structural language through which raw engineering data is represented, compressed, and interpreted, with particular emphasis on Principal Component Analysis for dimensionality reduction. Probabilistic reasoning and Bayesian inference are discussed as the basis for modelling measurement uncertainty and for deriving commonly used loss functions from statistical first principles. Optimisation theory, including gradient-based, constrained, and non-convex methods, is presented as the computational mechanism through which model parameters are learned from data. A unifying case illustration drawn from structural health monitoring demonstrates how these three mathematical domains interact within a single engineering learning pipeline, from sensor data acquisition through feature representation to model training. The chapter concludes by identifying open mathematical challenges in non-convex optimisation, high-dimensional inference, and physics-informed learning that continue to shape ongoing research in artificial intelligence for engineering applications, and argues that mathematical fluency remains essential for the responsible design and deployment of learning-based engineering systems. The discussion deliberately avoids treating machine learning as a purely computational discipline, positioning it instead within the broader tradition of applied mathematics, where abstract theory and practical engineering problem solving inform one another. Particular attention is given to how classical mathematical results, developed long before the current wave of artificial intelligence research, continue to underpin state-of-the-art learning algorithms, illustrating the enduring relevance of foundational mathematics education for engineering practitioners and researchers alike.
Authors: T. Maruthi, I. Paulraj Jayasimman
Institution: AMET University
Language: English
Subfield: Artificial Intelligence
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7204073374