SoSe 22: High-dimensional probability
Péter Koltai
Additional information / Pre-requisites
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Comments
This seminar is about random objects in very high-dimensional Euclidean spaces, as they appear for instance in data science. One intriguing property of the associated theory is that our low-dimensional intuition often fails spectacularly, as the following examples show.
- An n-dimensional standard normal distribution is highly concentrated around a sphere (i.e., the surface of a ball) of radius square root of n, rendering our „bell-curve“ intuition useless.
- Let us consider the unit sphere in hight dimensions, and let A be a subset of it covering at least half of its area. Then any small neighborhood of A will be exponentially close (with respect to the size of the neighborhood) to covering the whole sphere.
We will explore the mathematical background of these phenomena and review, in particular, inequalities concerning concentration of probability. Depending on number of participants and interest we might even learn about their consequences for random graphs and matrix completion.
Prerequisites:
A rigorous course in probability theory, further undergraduate linear algebra and calculus.
Literature:
R. Vershynin: High-Dimensional Probability
https://www.math.uci.edu/~rvershyn/papers/HDP-book/HDP-book.pdf
Suggested reading
R. Vershynin: High-Dimensional Probability
https://www.math.uci.edu/~rvershyn/papers/HDP-book/HDP-book.pdf
14 Class schedule
Regular appointments