Fall 2025: Random Matrix Theory in Data Science and Statistics (EN.553.796)

Course Description

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

  • Geometric method of analysis with concentration inequalities
  • Invariance and relation to random projection
  • Application: Johnson-Lindenstrauss transform and dimensionality reduction
  • Application: Compressed sensing

2. Classical theory of i.i.d. random matrices

  • Moment method for eigenvalue limit theorems
  • Semicircle law for Wigner matrices
  • Marchenko-Pastur law for Wishart matrices
  • Elements of universality
  • Elements of free probability theory
  • Application: Neural networks and random optimization landscapes
  • Application: Covariance estimation

3. Spiked matrix models and principal component analysis (PCA)

  • Resolvent method for eigenvalue limit theorems and further applications to eigenvectors
  • Baik—Ben Arous—Péché (BBP) phase transition in the performance of PCA
  • Computational challenges and statistical-computational gaps in PCA
  • Application: Non-linear improvements to PCA
  • Application: Community detection

4. Matrix concentration inequalities

  • General concentration inequalities and applications to random matrix norms (bounded differences, Lipschitz concentration, etc.)
  • General-purpose bounds on expected random matrix norms (matrix Chernoff, Bernstein, etc.)
  • Non-commutative Khintchine inequality and its exciting recent extensions
  • Application: Randomized numerical linear algebra
Contact & Office Hours

I am the instructor of this course, Tim Kunisky, and the teaching assistant is AMS PhD student Yue Wu.

The best way to contact us is by email, at kunisky [at] jhu.edu and ywu166 [at] jhu.edu, respectively. We will conduct a survey in the first week or two of class to pick a time for office hours.

Schedule

Class will meet Tuesdays and Thursdays, 12:00pm to 1:15pm in Bloomberg 176.

Below is a tentative schedule, to be updated as the semester progresses.

Date Details
Week 1
Aug 26 1. Course logistics. Random vector theory. Properties of Gaussian random vectors. Concentration inequalities. Consequences for how to think about high-dimensional geometry.
Aug 28 2. PCA and SVD as exploratory data analysis tools. Eckart-Young-Mirsky theorem. Multiplying by a Gaussian matrix: what does it do? What is it good for? How to think about Gaussian processes.
Week 2
Sep 2 3. Random matrices for dimensionality reduction: the Johnson-Lindenstrauss transform and lemma. "First moment method" proof technique using union bounds.
Sep 4 4. Concentration of singular values of short wide matrices. Interpretation as a "matrix concentration" inequality. "Geometric method" for random matrix analysis. Discretizing matrix norms with epsilon nets.
Lecture Notes and Materials

You do not need to buy any books for this course. I will post lecture notes here as the course goes on. If you want to look ahead to future topics, you can look at last year's notes here, but be advised that the topics covered this year might differ slightly. If you notice typos in either set of 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:

Grading

Grades will be based on a small number of written homework assignments, class participation, and a final project concerning a recent research paper, open problem, or topic of interest related to the material we cover.

Assignments

Homework will be posted here, and is to be submitted through Gradescope (details will be announced on Canvas soon). Please try to talk to me in advance if you need more time for an assignment.

Final Project

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 two parts:

  • A short in-class presentation during the final exam period.
  • A short written report, also due during the final exam period.

Further details on both, as well as a list of reasonable choices of topics and references for the project, will be posted during the first few weeks of class.