Course period: august to december, 2026.
General Information
Professor: Guilherme Ost
My office: 414
Class schedule and location: Mondays and Wednesdays, 9:00–10:30 a.m., in Room 236.
Course objective and syllabus
The goal of this course is to develop probabilistic tools that allow us to tackle statistical problems in which the dimension of the problem is high, relative to the number of observations at our disposal. The course focuses mainly on theory, though we will try to illustrate the discussed results through numerical examples.
The tentative syllabus of the course is given in the list below:
Basic results on concentration of measure.
Uniform laws of large numbers.
Metric entropy and applications.
Random matrices and covariance matrix estimation.
Sparse linear models in high dimensions.
Principal component analysis in high dimensions.
Regularized M-estimators.
Matrix estimation with rank constraints.
Minimax lower bounds.
Central limit theorems in high dimensions and applications.
Prerequisites
It is recommended that students have already taken a course in analysis, linear algebra and probability. On the other hand, although having already taken a statistics course helps, I would say that this is not essential for understanding the course material.
Students who have not taken one of the courses recommended above may nevertheless still be able to follow the course. However, the material presented will very likely be harder to digest.
References:
We will follow the lecture notes made available here. They will follow very closely Chapters 1 to 10 and 15 of the book
WAINWRIGHT, M. - High-Dimensional Statistics: A Non-Asymptotic Viewpoint (Cambridge Series in Statistical and Probabilistic Mathematics), 2019.
Other good references that will occasionally be used are the books
VERSHYNIN, R. - High-Dimensional Probability: An Introduction with Applications in Data Science (Cambridge Series in Statistical and Probabilistic Mathematics), 2018.
GIRAUD, C. - Introduction to High-Dimensional Statistics (Chapman & Hall/CRC Monographs on Statistics and Applied Probability, 139), 2015.
At this link, the interested student will find a YouTube series of 6 videos by Christophe Giraud on High-Dimensional Statistics, based on his book.
Evalutation criteria
To be defined depending on the number of students enrolled in the course. The details will be discussed on the first day of class and will subsequently be posted here.