High-Dimensional Probability with Applications to Data Science

Data sciences play an increasingly prominent role in modern society and are developing quickly. Probabilistic methods often provide foundation and inspiration for such developments. Particularly in the much-discussed regime of "big data", the methods draw upon the elegant mathematics of high- and infinite-dimensional probability. Building upon the probability and linear algebra from basic undergraduate courses, this course will cover the key probabilistic methods and results that form an essential toolbox for a mathematical data scientist.

We will follow the draft lecture notes of Roman Vershynin, "High-Dimensional Probability: An Introduction with Applications in Data Science", 2017, which can be found on the internet. The seminar meetings will summarise sections of the lecture notes. Students taking the course for credit will be required to present one or more sections in class (minimum of one, with additional credit for multiple presentations).

Topics:

• Preliminaries on random variables
• Concentration of sums of independent random variables
• Random vectors in high dimensions
• Sub-Gaussian random matrices
• Concentration without independence
• Quadratic forms, symmetrisation, and contraction
• Random processes
• Chaining
• Deviations of random matrices and geometric consequences
• Sparse recovery and compressed sensing

• Bühlmann, Peter; van de Geer, Sara. Statistics for High-Dimensional Data. Methods, Theory and Applications. Springer Series in Statistics. Springer, Heidelberg, 2011. xviii+556 pp. ISBN: 978-3-642-20191-2
• Kaipio, Jari; Somersalo, Erkki. Statistical and Computational Inverse Problems. Applied Mathematical Sciences, 160. Springer-Verlag, New York, 2005. xvi+339 pp. ISBN: 0-387-22073-9
• Reich, Sebastian; Cotter, Colin. Probabilistic Forecasting and Bayesian Data Assimilation. Cambridge University Press, New York, 2015. x+297 pp. ISBN: 978-1-107-66391-6; 978-1-107-06939-8
• Smith, Ralph C. Uncertainty Quantification. Theory, implementation, and applications. Computational Science & Engineering, 12. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2014. xviii+382 pp. ISBN: 978-1-611973-21-1
• Sullivan, T. J. Introduction to Uncertainty Quantification. Texts in Applied Mathematics, 63. Springer, Cham, 2015. xii+342 pp. ISBN: 978-3-319-23394-9; 978-3-319-23395-6