High-Dimensional Probability
Geometry, concentration, random matrices, and stochastic processes
High-dimensional probability studies random objects whose ambient dimension is large enough that ordinary geometric intuition becomes unreliable. Volume migrates toward boundaries, naïve discretizations become impossibly large, and yet randomness often produces remarkably stable behavior.
The organizing question. Which notions of size and complexity continue to be useful when dimension is large, and how do they control the behavior of random vectors, matrices, and processes?
The notes answer this question in stages. The first lectures measure sets by covering numbers and by Gaussian width, and show that the two disagree in high dimension. The middle lectures build the probabilistic toolkit, from sub-Gaussian tails through nets for random matrices to Gaussian processes, concentration, and comparison inequalities. The final lectures prove that Gaussian width is, up to constants, the expected supremum of every Gaussian process, through chaining and Talagrand’s majorizing measure theorem, and use that fact to determine how many samples estimate a covariance matrix and how many random measurements recover a sparse signal. The recurring lesson is that the right measure of the size of a set is its width, not its dimension and not its volume.
These notes support MATH 598 / 784 at McGill University in Fall 2026.
Themes
High-dimensional geometry and basic probabilistic tools
The curse of dimensionality, convex hulls and sparse approximation, Gaussian width, and tails of random variables.
Concentration, embeddings, and random matrices
Orlicz norms, Bernstein’s inequality and dimension reduction, coordinate embeddings, extreme singular values, and covariance estimation.
Gaussian processes
Gaussian processes and the isonormal process, Gaussian concentration, Gaussian comparison and Gaussian matrices, and Gaussian width revisited.
Convex relaxations
Semidefinite relaxations and max-cut and Grothendieck’s inequality.
Chaining and applications
Dudley’s entropy integral, applications of Dudley’s inequality, generic chaining, the lower bound and the matrix deviation inequality, covariance estimation for low-dimensional data, and sparse recovery.
How to read these notes
Each lecture is a self-contained unit written for a reader who has the earlier lectures at hand. Later lectures cite earlier results by lecture number rather than by link, so a phrase such as “the net lemma of Lecture 8” is a pointer to that page.
- Proofs are collapsed by default. Every theorem, lemma, and proposition is followed by an expandable proof. Read the statements first and open the proofs when you want the argument; in print, all proofs are expanded.
- Asides are collapsed too. Remarks, examples, and digressions that are not needed for the main line of a lecture sit in expandable boxes with a descriptive title.
- Constants. The letters \(c\) and \(C\) denote absolute constants whose values may change from line to line. A random variable is called \(K\)-sub-Gaussian when \(K\) is a variance proxy, so that constants in later results depend on \(K\) rather than on \(K^2\); the conventions are fixed in Lectures 4 and 5.
- Prerequisites. A first graduate course in probability and comfort with norms, convexity, and linear algebra. Measure-theoretic details are kept to a minimum and flagged where they matter.
Sources and acknowledgements
The selection and organization of the course owe a particular debt to four resources:
- Roman Vershynin, High-Dimensional Probability: An Introduction with Applications in Data Science (Vershynin 2026);
- Ramon van Handel, Probability in High Dimension (Handel 2016);
- Shahar Mendelson, Probabilistic Methods in Analysis (Mendelson 2023); and
- Michel Talagrand, Upper and Lower Bounds for Stochastic Processes (Talagrand 2014), for the chaining lectures.
The notes synthesize ideas from these sources with material developed in earlier versions of the course. Any errors in this presentation are, of course, mine.
Fall 2026
Course logistics live separately from the durable notes. See the course page, the date-by-date calendar, and the exercise sheets.
Download Exercise Sheet 4 (PDF), for discussion Friday, October 2. The sheet covers sampling and covering numbers, randomized Hadamard coordinate embeddings, quadratic forms on nets, and norm deviations on nets.