This chapter presents the matrix deviation inequality, a uniform deviation bound for random matrices over general sets. Applications include two-sided bounds for random matrices, refined estimates for random projections, covariance estimation in low dimensions, and an extension of the Johnson–Lindenstrauss lemma to infinite sets. We prove two geometric results: the M* bound, which shows how random slicing shrinks high-dimensional sets, and the escape theorem, which shows how slicing can completely miss them. These tools are applied to a fundamental data science task – learning structured high-dimensional linear models. We extend the matrix deviation inequality to arbitrary norms and use it to strengthen the Chevet inequality and derive the Dvoretzky– Milman theorem, which states that random low-dimensional projections of high-dimensional sets appear nearly round. Exercises cover matrix and process-level deviation bounds, high-dimensional estimation techniques such as the Lasso for sparse regression, the Garnaev–Gluskin theorem on random slicing of the cross-polytope, and general-norm extensions of the Johnson–Lindenstrauss lemma.

This chapter develops a non-asymptotic theory of random matrices. It starts with a quick refresher on linear algebra, including the perturbation theory for matrices and featuring a short proof of the Davis–Kahan inequality. Three key concepts are introduced – nets, covering numbers, and packing numbers – and linked to volume and error-correcting codes. Bounds on the operator norm and singular values of random matrices are established. Three applications are given: community detection in networks, covariance estimation, and spectral clustering. Exercises explore the power method to compute the top singular value, the Schur bound on the operator norm, Hermitian dilation,Walsh matrices, the Wedin theorem on matrix perturbations, a semidefinite relaxation of the cut norm, the volume of high-dimensional balls, and Gaussian mixture models.

This chapter explores methods of concentration that do not rely on independence. We introduce the isoperimetric approach and discuss concentration inequalities across a variety of metric measure spaces – including the sphere, Gaussian space, discrete and continuous cubes, the symmetric group, Riemannian manifolds, and the Grassmannian. As an application, we derive the Johnson–Lindenstrauss lemma, a fundamental result in dimensionality reduction for high-dimensional data. We then develop matrix concentration inequalities, with an emphasis on the matrix Bernstein inequality, which extends the classical Bernstein inequality to random matrices. Applications include community detection in sparse networks and covariance estimation for heavy-tailed distributions. Exercises explore binary dimension reduction, matrix calculus, additional matrix concentration results, and matrix sketching.