Fundamen Matemat p Cs de Datos

Mathematical Foundations of Data Science

Class notes, announcements, and other information can be found here.

Instructor: Anastasios Matzavinos, amatzavinos@uc.cl 

Teaching assistants: Kai Yamamoto Kalm and Ignacio Vergara Briones

Class meeting times:  Tuesday & Thursday 12:20 pm - 1:30 pm in room B25.

TA office hours: Wednesday 9:40 am - 10:50 am (Kai) and Thursday 3:00 pm - 4:00 pm (Ignacio).

Instructor office hours: By appointment.

Course description: IMT3120 provides a mathematically rigorous introduction to many of the statistical and machine learning methods that are central to modern data science. The course is organized around recent developments in nonparametric and high-dimensional statistical inference. After reviewing the fundamentals of probability, ordinary linear regression, and classical hypothesis testing, we will study a collection of methods that have become standard in contemporary data science, including ridge regression, the LASSO, kernel density estimation, spline methods, large-scale kernel machines, and graphical models for high-dimensional data. Time permitting, we will also discuss selected topics in the mathematical theory of neural network approximation.

While the course introduces a variety of statistical methods and machine learning algorithms, its primary emphasis is on developing a rigorous understanding of their underlying theory and establishing mathematical guarantees for their behavior. To this end, we will develop a number of fundamental tools from modern probability, statistics, and analysis, including concentration inequalities, random matrix theory, reproducing kernel Hilbert spaces, Sobolev space methods for nonparametric estimation, Vapnik–Chervonenkis theory, and uniform laws of large numbers. By the end of the course, students will have acquired a rigorous mathematical framework for understanding, analyzing, and extending many of the core methods that underpin contemporary data science.

References: The following references will be used in different parts of the course. 

• R. Samworth and R. Shah. Modern Statistical Methods and Theory: An Introduction to Nonparametric and High-Dimensional Statistics. Cambridge University Press, 2026.
• K. Spiliopoulos, R. Sowers, and J. Sirignano. Mathematical Foundations of Deep Learning Models and Algorithms. American Mathematical Society, 2025.
• A. Tsybakov. Introduction to Nonparametric Estimation. Springer, 2009.
• R. Vershynin. High-Dimensional Probability: An Introduction with Applications in Data Science. 2^nd Edition. Cambridge University Press, 2026.

Grading policy: The final grade will be based on attendance (5% of the grade), homework assignments (35%), a midterm exam (30%),  and a final take-home exam (30%).

Homework assignments: Homework problems will be handed out on a regular basis. Discussion of homework assignments with other students is encouraged, but what is handed in should be your own work. 

Announcements and other information about the class can be found here. A PDF copy of the syllabus can be found here: IMT_3120.pdf