Foundations of Machine Learning and AI [Singh]

[mklabgr]

Foundations of Machine Learning and AI: Geometry, Probability and Optimization
Authors: Pradeep Singh, Balasubramanian Raman
Publisher: Springer Cham
Series:Studies in Big Data, Vol. 199
Publication: eBook — 17 August 2026; Hardcover — 18 August 2026
Edition: 1st
Pages: XXX + 558
DOI: 10.1007/978-3-032-30336-3
Subjects: Machine Learning, Artificial Intelligence, Data Science, Linear Algebra, Probability, Statistics 

Summary
Foundations of Machine Learning and AI is a mathematically oriented textbook designed to build the theoretical foundations needed to understand modern machine learning rather than simply use ML libraries as black boxes. Its central idea is that two mathematical viewpoints underpin much of contemporary AI: geometry, mainly through linear algebra, and uncertainty, through probability and statistics. The authors develop these themes together, showing how concepts such as inner products, projections, eigenvalues, singular-value decomposition, probability distributions, estimation and generalization lead naturally to practical machine-learning algorithms. 

The first half develops the necessary linear algebra and geometry: Euclidean spaces $\mathbb{R}^n$, norms, inner products, projections, linear transformations, matrices, spectral theory, SVD, PCA and numerical linear algebra. Importantly, these subjects are connected directly with ML applications—for example, PCA for denoising and dimensionality reduction, matrix conditioning in regression, and randomized/sketching methods for solving very large least-squares problems. 

The second half moves into probability, statistics and optimization. It covers random variables, expectation, moments and concentration inequalities, then statistical estimation through MLE, MAP and Bayesian methods, hypothesis testing and resampling. These foundations lead to regression and classification, kernels, dimensionality reduction and techniques for high-dimensional data. The final chapter develops optimization methods specifically for statistical learning. The overall progression is therefore approximately
$$
\text{Linear Algebra}
\rightarrow
\text{Geometry}
\rightarrow
\text{Probability}
\rightarrow
\text{Statistics}
\rightarrow
\text{Machine Learning}
\rightarrow
\text{Optimization}.
$$

The emphasis throughout is on deriving algorithms from first principles and understanding why they work, rather than merely presenting recipes. Springer explicitly positions examples around real ML problems such as regularization selection, PCA denoising, regression instability and scalable least-squares computation. 

Key takeaways

• Strong mathematical foundation for ML: it connects linear algebra, probability, statistics and optimization into one continuous treatment rather than teaching them as unrelated subjects.
  
• Geometry plays a central role: projections, eigenvectors, spectra, SVD and PCA provide a geometric interpretation of many machine-learning methods.
  
• Probability explains uncertainty and learning: estimation, concentration and generalization provide the theoretical basis for understanding what a model can infer from finite data.
  
• Theory is tied to algorithms: the book repeatedly connects mathematical results to regression, classification, kernels, dimensionality reduction and large-scale computation. 
  

BOOK