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Random Matrix Theory (MA5346) - SS 15

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LecturerFelix Krahmerfelix.krahmeremattum.de MI 02.10.033
TutorFelix Krahmerfelix.krahmeremattum.de MI 02.10.033
LectureThu. 10:15 - 11:45MI 02.08.020
TutorialWed. 14:15 - 15:45MI 02.08.020

News

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Content of the Lecture

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A random matrix A is a matrix-valued random variable. For example, the entries may be i.i.d. scalar Gaussian or Bernoulli random variables, but random matrices with dependent entries will also be considered. There are many classical results about the asymptotic behaviour of the spectrum of such matrices, like for example Wigner's semicircle law. However, it has recently become important to also understand the non-asymptotic spectral behaviour of random matrices. A typical question of interest is for example the following: Consider random matrices A of size m x N, how much do the singular values of their realizations differ from the predicted asymptotic behaviour?

The class will, in large parts, follow the lecture notes by Roman Vershynin [1].

We will start with non-asymptotic deviation estimates for random variables in one dimension. We introduce subgaussian and sub exponential random variables and random vectors as well as the isotropic random vectors. The concepts then appear in the study of random matrices, mainly matrices with independent rows or columns. We will prove results on the tail behaviour of their maximal singular values for different types of distributions for the row/column vectors. Furthermore various applications will be discussed, including dimension reduction [2] and compressed sensing [3].

Exercise Sheets

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Lecture Recording

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The recording of the lecture will be put online as soon as possible after the lecture. Due to technical problems it can happen that the audio was not recorded for short periods of time (e.g. beginning of Lecture 2). Note that the recording of the lecture is a special service, however we are not able to assure that each lecture is recorded. When you click on the respective link a download of the MP4 file is started. For further technical problems contact Steffen Peter. Please download the respective lecture video file(s) as soon as possible after the lecture took place. We are going to remove the files about 7 - 14 days after we put it online. If you missed to download it, please also contact Steffen Peter.

Lecture 1 (Apr 16): not recorded
Lecture 2 (Apr 22): please contact us
Lecture 3 (Apr 23): please contact us
Lecture 4 (Apr 30): please contact us
Lecture 5 (May 6): please contact us
Lecture 6 (June 3): please contact us
Lecture 7 (June 10): please contact us
Lecture 8 (June 11): please contact us
Lecture 9 (June 17): please contact us
Lecture 10 (June 24): please contact us
Lecture 11 (June 25): please contact us
Lecture 12 (July 1): Part 1 Part 2 Part 3 Part 4
Lecture 13 (July 8): Part 1 Part 2 Part 3  

Literature

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[1] Vershynin, R.: Introduction to the non-asymptotic analysis of random matrices, in Compressed Sensing, Theory and Applications, ed. Y. Eldar and G. Kutyniok. Cambridge University Press, 2012, pp. 210-268 [.pdf Pfeil]
[2] Krahmer, F. and Ward, R.: New and improved Johnson-Lindenstrauss embeddings via the Restricted Isometry Property, SIAM J. Math. Anal. 43(3), 2011, 1269-1281.
[3] Foucart, S. and Rauhut, H.: A Mathematical Introduction to Compressive Sensing, Applied and Numerical Harmonic Analysis, Birkhäuser, 2013
[4] Tropp, J.: User-friendly tail bounds for sums of random matrices, Found. Comp. Math. 12(4), 2012, 389-434.

-- FelixKrahmer - 22 Apr 2015