This is a first course in random matrix theory, the study of the eigenvalues and eigenvectors of matrices with random entries that is foundational to high-dimensional statistics and data science. Aside from the main ideas and modern applications of random matrices, a key goal will be to introduce you to the main concepts of probability in high dimensions: concentration of measure, the geometry of high-dimensional spaces and convex sets, Gaussian measure, and sharp transitions and threshold phenomena. The following is a (very) tentative ordered list of specific topics to be covered:
1. Gaussian matrices and dimensionality reduction
2. Classical theory of i.i.d. random matrices
3. Spiked matrix models
4. Matrix concentration inequalities
Prerequisites: Linear algebra and either Real Analysis (AS.110.405 or equivalent) or Probability Theory I (EN.553.720 or equivalent). You should be familiar with eigenvalues, eigenvectors, and singular value decompositions; formal definitions of limits, convergence, and open and closed sets; laws of large numbers and central limit theorems; and conditional expectation. Deep background in measure-theoretic probability is not necessary, but some topics may be difficult if you are completely unfamiliar with it.
I am the instructor of this course, Tim Kunisky, and the teaching assistant is AMS PhD student Zhouhao Yang.
The best way to contact us is by email, at kunisky [at] jhu.edu and zyang145 [at] jhu.edu, respectively. We will choose regular office-hour times after the first week of class. In the meantime, please contact us directly to schedule an appointment.
Class meets Tuesdays and Thursdays, 9:00am to 10:15am in Maryland 104.
Below is a tentative schedule, to be updated as the semester progresses.
| Date | Details |
|---|---|
| Week 1 | |
| Sep 1 | 1. Course logistics. Random vector theory. Concentration inequalities and high-dimensional geometry. Properties of Gaussian random vectors. |
| Sep 3 | 2. Orthogonal invariance, Gaussian random vectors, and uniform measure on the sphere. Multiplication by Gaussian random matrices. Johnson-Lindenstrauss lemma and dimensionality reduction. "First moment method" proof technique. |
| Week 2 | |
| Sep 8 | 3. Finish proof and discussion of Johnson-Lindenstrauss lemma. Extensions and sketch of applications in machine learning. Introduction to "geometric method" for singular values of rectangular matrices. |
| Sep 10 | 4. Concentration of singular values of short fat matrices. Interpretation as a "matrix concentration" inequality. Discretizing matrix norms with epsilon nets. Non-constructive proof of existence of good epsilon nets. |
You do not need to buy any books for this course. I will post updated lecture notes here as the course proceeds. In the meantime, you can consult the Fall 2025 lecture notes and the Fall 2024 lecture notes, but the topics and presentation may differ this year. If you notice typos in any of the notes, please let me know.
The following are books or lecture notes that cover some similar material and might be useful to you in addition to my notes. These are useful references on general advanced probability; the first is especially friendly:
These are additional references on random matrix theory specifically:
Grades will be based on a small number of written homework assignments (25%), a written in-class midterm exam (25%), and a final project proposal (5%), report (10%), and presentation (35%) concerning a recent research paper, open problem, or topic of interest related to the material we cover.
The midterm exam will not ask you to solve new problems on the spot. Instead, you will be asked to write up solutions to a selected previous homework problem and to outline the proof to a selected major theorem we have discussed in class.
The following are the policies for submitted work in this course:
Homework will be posted here and submitted through Gradescope (see Canvas announcements for details). Please talk to me in advance if you need more time for an assignment.
| Assigned | Due | Link |
|---|---|---|
| Assignment dates and links to be added. | ||
Your final project is to do one of the following on a topic related to the content of this course: (1) read and digest a paper and present its content in your own words and style, with some elaboration that is not present in the original paper; (2) perform an interesting computational experiment motivated by something we have seen in class or something you read in a paper and report in detail on the results and their interpretation; or (3) for the intrepid, find an open problem related to something we have seen in class, try to work on it, and report your findings.
The project submission will have three parts:
More detailed requirements, due date for the proposal, and suggestions of reasonable project topics will be posted later.