High-Dimensional Statistics - LTCC
Graduate course, LTCC, London, 2025
Five week introductory course on high-dimensional statistics. Course notes and slides will be posted here and updated on a rolling basis. The course material is a simple convex combination of available materials from existing books and notes from Philippe Rigollet, Jan-Christian Hütter, Roman Vershynin and Martin Wainwright, all of which will be linked below. I would consult these for a more complete picture.
Annotated slides Week 1 Week 2 Week 3 Week 4 Week 5
Lecture notes (last update 4th of March, will try to update weekly) Lecture notes
Resources High-Dimensional Statistics by Philippe Rigollet and Jan-Christian Hütter: Link
High-Dimensional Statistics: A Non-Asymptotic Viewpoint by Martin Wainwright: Link
High-Dimensional Probability by Roman Vershynin: Link (there is a free draft version of the book available on his website)
Evaluation
For the assignment/evaluation component of the course please take a look a at the following list of papers and write a 6-10 page report excluding references with roughly the following format:
An introduction to the context of the paper, i.e. why is it important to show the results that they aim to show. If the paper you chose is a review paper, then explain why it is necessary to have all these results collected and what it adds to the literature.
Explanation of what you think are the main results of the paper, and discuss the assumptions that are needed to show this result. If the paper is a review paper, then you can select a few of the results and discuss them in a similar manner.
Identification of some “main” technical/mathematical difficulties in the proofs of the discussed results in point 2. –what are the “hard/novel” parts of the proofs? If it is a review paper, and there are no proofs involved then you can skip this part, but you can add more to point 4.
Discussion of how the results of the paper are being used in other works, an easy way to do this is to see who cited the paper and see how they used results from it AND/OR you can look for extensions of the results presented.
You can deviate away from this format if you think that your chosen paper is not suitable for it. If you can’t find anything interesting on the list of papers, you can look around for something that fits with your interest and contains some elements of high-dimensional statistics; you can ask me first if you are not sure that your chosen paper would fit the theme of the course.
List of papers
Bellec, P.C., Lecué, G. and Tsybakov, A.B., 2018. Slope meets lasso: improved oracle bounds and optimality. The Annals of Statistics, 46(6B), 3603-3642.
Johnstone, I. M. 2001. On the distribution of the largest eigenvalue in principal components analysis. Annals of Statistics, 29(2), 295–327.
Massart, P. 1990. The tight constant in the Dvoretzky-Kiefer-Wolfowitz inequality. Annals of Probability, 18, 1269–1283.
Ravikumar, P., Wainwright, M. J., Raskutti, G., and Yu, B. 2011. High-dimensional covariance estimation by minimizing 1-penalized log-determinant divergence. Electronic Journal of Statistics, 5, 935–980.
van de Geer, S., and Buhlmann, P. 2009. On the conditions used to prove oracle results for the Lasso. ¨ Electronic Journal of Statistics, 3, 1360–1392.
E. Abbe, A. S. Bandeira, G. Hall. Exact recovery in the stochastic block model, IEEE Transactions on Information Theory 62 (2016), 471–487.
O. Guedon, R. Vershynin, Community detection in sparse networks via Grothendieck’s inequality, Probability Theory and Related Fields 165 (2016), 1025–1049.
M. Talagrand, A new look at independence, Ann. Probab. 24 (1996), 1–34.
Y. Yu, T. Wang, R.J. Samworth, A useful variant of the Davis-Kahan theorem for statisticians, Biometrika 102 (2015), 315–323.
Cutting, Christine, Davy Paindaveine, and Thomas Verdebout. Testing uniformity on high-dimensional spheres against monotone rotationally symmetric alternatives. The Annals of Statistics 45 (2017): 1024-1058.