Mathematical Pathways to Machine Learning: Calculus, Linear Algebra, and Optimization by Xiantao Li is an introductory textbook designed to bridge the gap between foundational mathematics and machine learning. Developed from Penn State’s MATH 225 course, it is aimed primarily at first-year and early-sophomore students who have completed one semester of calculus and have some programming experience, but no prior course in linear algebra, multivariable calculus, or optimization.
The book presents mathematics not as separate subjects but as an interconnected toolkit for understanding machine learning: linear algebra provides the language for representing data, calculus describes how models change and how backpropagation works, and optimization turns these mathematical ideas into learning algorithms.
The 15-week structure is divided into three major parts. Linear algebra covers vectors, matrices, projections, eigenvalues and eigenvectors, SVD, and PCA, with applications such as PageRank. Vector calculus and optimization introduce gradients, Jacobians, Hessians, the chain rule, backpropagation, Taylor approximations, least squares, and regression.
Finally, iterative and constrained optimization develops gradient descent, momentum, Newton’s method, Lagrange multipliers, KKT conditions, duality, regularization, and support-vector machines. Throughout, the author emphasizes geometric intuition, hand calculations, worked examples, and connections to real machine-learning models, with computational exercises designed around NumPy.
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The book presents mathematics not as separate subjects but as an interconnected toolkit for understanding machine learning: linear algebra provides the language for representing data, calculus describes how models change and how backpropagation works, and optimization turns these mathematical ideas into learning algorithms.
The 15-week structure is divided into three major parts. Linear algebra covers vectors, matrices, projections, eigenvalues and eigenvectors, SVD, and PCA, with applications such as PageRank. Vector calculus and optimization introduce gradients, Jacobians, Hessians, the chain rule, backpropagation, Taylor approximations, least squares, and regression.
Finally, iterative and constrained optimization develops gradient descent, momentum, Newton’s method, Lagrange multipliers, KKT conditions, duality, regularization, and support-vector machines. Throughout, the author emphasizes geometric intuition, hand calculations, worked examples, and connections to real machine-learning models, with computational exercises designed around NumPy.
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