Mathematics for Data Science

The goal of this course is to provide the mathematical foundations underlying several core methods in modern Data Science, Machine Learning, and Quantitative Finance. In particular, the course aims to develop the tools required to understand convex optimization, which plays a central role in model estimation, loss-function design, and decision-making problems, as well as kernel methods, which form the basis of many modern learning algorithms.

To achieve this goal, the course first introduces the mathematical structures arising in linear algebra and functional analysis, emphasizing both theoretical rigor and geometric intuition. Topics include vector spaces, linear operators, normed spaces, inner-product spaces, Banach spaces, and, ultimately, Hilbert spaces. Building on these foundations, the course studies convex optimization and its applications to estimation and learning problems, as well as reproducing kernel Hilbert spaces (RKHS), the representer theorem, and their classical applications, including kernel ridge regression and support vector machines.

Prerequisites: Familiarity with single and multivariable calculus, linear algebra, and probability.

By the end of the course, students will be able to:

• Understand the mathematical foundations of linear algebra, functional analysis, and convexity that provide the basis for modern data science methods.

• Formulate and analyze optimization and learning problems using the language of vector spaces, Hilbert spaces, and convex optimization.

• Understand the theoretical principles behind kernel methods and apply them to classical machine learning tasks such as regression and classification.

In presenza

The course is structured through alternating theory and practice sessions. Students will solve take-home exams and quizzes.

The final course assessment is an open-book final written exam.