Applied Mathematics

The course covers key concepts and methods in high-dimensional probability and statistics, with a focus on concentration inequalities, subgaussian and subexponential random variables, and their applications in data analysis and machine learning. Topics include: Hoeffding inequality and the definition of subgaussian random variables. Equivalent characterizations and the subgaussian norm. Hoeffding inequality for sums of independent subgaussian variables. Subexponential random variables and Chernoff bounds. Johnson–Lindenstrauss lemma and dimensionality reduction. Epsilon-nets, covering numbers, and packing numbers; quantitative bounds in 𝑅 𝑛 R n . Concentration of the operator norm of random matrices with subgaussian entries and related spectral results (Courant–Fischer, Weyl, Davis–Kahn theorems). Stochastic Block Model and community detection problems. Spectral clustering algorithm and k-means algorithm. Laplacian operator and unnormalized Laplacian spectral clustering. Strong law of large numbers, Glivenko–Cantelli theorem, and the Dvoretzky–Kiefer–Wolfowitz (DKW) inequality. VC dimension, Rademacher complexity, and their role in statistical learning theory. Throughout the course, both theoretical and computational exercises are assigned and discussed, some of which are implemented during the laboratory sessions.

A basic knowledge of probability and statistics is required, including elementary concentration inequalities, discrete and continuous random variables, expectation and variance, independence, the law of large numbers, and the central limit theorem. Familiarity with linear algebra (eigenvalues, eigenvectors, matrix norms) and basic mathematical analysis (sequences, limits, derivatives) is also useful. Knowledge of the Matlab programming language is required, as it will be used during the laboratory sessions to perform the assigned computational tasks.

R. Vershynin. ``High-Dimensional Probability. An Introduction with Applications in Data Science". Cambridge University Press. Disponibile online (gratuitamente). M. J. Wainwright. ``High-Dimensional Statistics. A Non-Asymptotic Viewpoint". Cambridge University Press.

Lectures will be delivered in presence. Attendance is strongly recommended.

Exams: oral and/or written exam.

R. Vershynin. ``High-Dimensional Probability. An Introduction with Applications in Data Science". Cambridge University Press. Disponibile online (gratuitamente). M. J. Wainwright. ``High-Dimensional Statistics. A Non-Asymptotic Viewpoint". Cambridge University Press.

The course includes theoretical lectures (for a total of 44 hours) delivered at the blackboard or using a tablet, focusing on probability exercises, theorem statements, and detailed proofs., In addition, 12 hours of laboratory sessions are scheduled, during which students use Matlab to carry out practical exercises directly related to the theoretical topics covered in class.