Math 439/549 - Topics in Probability Theory, Fall 2020


Content:
In this course we will study probabilistic methods in high dimensions and their applications to the following tentative list of problems: covariance estimation, community detection in networks, matrix completion, statistical learning theory, signal recovery. The theory part will include major concentration inequalities for both scalar- and matrix-valued random variables. The concentration inequalities are nonasymptotic counterparts of classical limit theorems that quantify how random variables deviate from some deterministic values. In the context of data science problems, they provide estimates of approximation errors and hold for any sample size.
 
Textbook:
R. Vershynin, High-Dimensional Probability: An Introduction with Applications in Data Science, Cambridge University Press, 2018, ISBN-10: 1108415199. [Available on the author's website.]
 
Review/study probability theory:
R. Durrett, Probability: Theory and Examples, Cambridge University Press, 5th edition, 2019, ISBN-10: 1108473687 (all topics). [Available on the author's website.]
S. Ross, A First Course in Probability, Pearson, 9th edition, 2012, ISBN-10: 032179477X (probability theory without measure theory)
Schaum's Outline of Probability and Statistics, McGraw-Hill Education, 4th edition, 2012, ISBN-10: 007179557X (outline of concepts with examples, except measure theory)
 
Course Mode: Remote Arranged
Lectures:Tue & Thur, posted on UNMLearn
Student Presentations:Scheduled, in Zoom
Questions/Answers:By email or appointment in Zoom. Reserved time is TR 2:00-3:15 pm.
Professor Info:Anna Skripka, skripka [at] math [dot] unm [dot] edu
 
Prerequisites:
The expected minimal prerequisites for the course are basic probability theory (at UNM, it is Math441/Stat461/Stat561), basic linear algebra, including singular value/eigenvalue decompositions of matrices, matrix norms, trace (at UNM, basic linear algebra is taught in Math 314 and Math 321 and basic matrix analysis in Math 464/514), and rigorous proof skills. Major prerequisite topics will be very briefly reviewed, but no class time will be dedicated to teaching them anew. A background in advanced calculus, real analysis, measure theory would be helpful.
 
Grades:
The course assessment is planned to be based on Zoom presentations.

Schedule

Aug 18, 20:  Review of probability basics, including textbook sections 1.1, 1.2.
Communication emails sent on 08/14, 08/19, 08/20. "Meet and greet" on Aug 18 at 2 pm if you are available.
 
Aug 25:  1.2 Introduction to concentration inequalities.
Aug 27:  Presentation "Vectors gone wild: A brief introduction to machine learning and data science" by Galen.
Communication email sent on 08/25.
 
Sep 1, 3:    1.3, 2.1 - 2.3 Review of limit theorems. Hoeffding's and Chernoff's inequalities.
Communication emails sent on 08/31 and 09/03.
 
Sep 8, 10:    2.5 Subgaussian distribution. 2.6 General Hoeffding's inequality.
Communication email sent on 09/08.
 
Sep 15, 17:    Boosting randomized algorithms. 2.7 Subexponential distribution. 2.8 Bernstein's inequality.
 
Sep 22, 24:    Review of basic matrix analysis, including textbook sections 4.1, 5.4.1, 5.4.2.
Communication email sent on 09/21.
 
Sep 29, Oct 1:    3.1 Random vectors. 3.2 Covariance matrices and principal component analysis.
 
Oct 6, 8:    4.4, 4.6 Norm and singular value bounds for matrices with subgaussian entries.
Communication email sent on 10/08.
 
Oct 13, 15:    5.4 Matrix Bernstein's and Khintchine's inequalities, their implications for applied problems. Symmetrization.
 
Oct 20, 22:    6.4 Derivation of symmetrization and other useful tricks. 6.5 Norms of random matrices with non-i.i.d. entries.
 
Oct 27, 29:    7.1-7.3 Random processes. Bounds for norms of Gaussian matrices. 7.5 Gaussian width. 9.4 M* bound. 8.1 Dudley's inequality. 8.3 VC dimension.
Communication email sent on 10/28.
 
Nov 2:    Presentation 4.5, 5.5 Community detection in networks (clustering)by Alexander.
Communication email sent on 11/02.
 
Nov 10:    Presentation 4.7 Covariance estimation for subgaussian distributions by Corbin.
Nov 12:    4.3.2 Error correcting codes.
Communication email sent on 11/11.
 
Nov 17:    Presentation 8.4 Statistical learning theory by Jonathan.
Nov 20:    Presentation 5.6 Covariance estimation for general distributions by Sushyam.
 
Nov 24:    Presentation 10.1, 10.2 Signal recovery by Sarah.
Communication email sent on 11/23.
 
Nov 30:    Presentation 6.6 Matrix completion by Stephen.
Dec 4:    Presentation 8.2.1, 8.2.2 Monte-Carlo Method and uniform law of large numbers by Rob.
Communication email sent on 12/03.
 
 
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