Math 549/Stat 569 - Topics in Probability Theory, Fall 2023
Time and Location:
Tue, Thur 2 - 3:15 pm, SMLC-352
Office Hours:
Tue, Thur, 11 - 11:50 am in SMLC 212 and also by appointment in Zoom
Professor Info:
Anna Skripka,
skripka [at] math [dot] unm [dot] edu
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.
[June 2020 version is available on the author's webpage.] Supplementary:
M. J. Wainwright, High-Dimensional Statistics: A Non-Asymptotic Viewpoint, Cambridge University Press, 2019, ISBN-10: 1108498027
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)
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 attendance and presentations.
Lecture notes:
Posted on UNM Canvas weekly.
Tentative SCHEDULE (to be updated at the end of each week)
Aug 22, 24:Review of probability basics, including textbook sections 1.1, 1.2. Basics of measure theory and Lebesgue integration.
Aug 29, 31:1.2, 1.3 Introduction to concentration inequalities. Review of limit theorems.
Sep 26, 28:4.5 Spectral analysis for community detection in networks. Review of basic matrix analysis, including 4.1, 4.5, 5.4.1, 5.4.2.
Oct 3, 5:3.1, 3.3, 3.4 Random vectors. 3.2 Covariance matrices and principal component analysis. Isotropic distributions.
Oct 10:Supplementary 1.2, 1.3 High-dimensional phenomena in statistics (covariance estimation, binary hypothesis testing).
Oct 17, 19:4.4, 4.5, 4.6 Norm and singular value bounds for matrices with subgaussian entries. Community detection.
Oct 24, 26:5.4 Tail and expectation bounds without independence of entries. Matrix Bernstein's inequality
Oct 31, Nov 2:5.4 Matrix Khintchine's inequality. 6.4 Symmetrization. 6.5 Norms of random matrices by symmetrization.
Nov 7, 9:7.1-7.3 Random processes. Bounds for norms of Gaussian matrices.
7.5 Gaussian width. 9.4 M* bound and escape theorem. 8.1 Dudley's inequality.
Nov 21:8.2.1, 8.2.2 Monte-Carlo method and uniform law of large numbers.
Nov 28, 30:4.7 Covariance estimation for subgaussian distributions. Clustering.
5.6 Covariance estimation for general distributions.
Dec 5, 7:10.5 Exact recovery and restricted isometry property.
Academic Integrity: Each student is expected to maintain the highest standards of honesty and integrity in academic and professional matters. The University reserves the right to take disciplinary action, including dismissal, against any student who is found responsible for academic dishonesty. Any student who has been judged to have engaged in academic dishonesty in course work may receive a reduced or failing grade for the work in question and/or for the course. Academic dishonesty includes, but is not limited to, dishonesty on quizzes, tests or assignments; claiming credit for work not done or done by others; and hindering the academic work of other students.
Learning the course material depends on completing and submitting your own work. UNM preserves and protects the integrity of the academic community through multiple policies including policies on student grievances (Faculty Handbook D175 and D176), academic dishonesty (FH D100), and respectful campus (FH CO9). These are in the Student Pathfinder and the Faculty Handbook.
Copyright Policy: All materials disseminated in class and on the course webpage are protected by copyright laws and are for personal use of students registered in the class. Redistribution or sale of any of these materials is strictly prohibited. Accommodations:
UNM is committed to providing equitable access to learning opportunities for students with documented disabilities. As your instructor, it is my objective to facilitate an inclusive classroom setting, in which students have full access and opportunity to participate. To engage in a confidential conversation about the process for requesting reasonable accommodations for this class and/or program, please contact Accessibility Resource Center at arcsrvs@unm.edu or by phone at 505-277-3506. Disclaimer: It is your responsibility to know and understand the course policies. The professor reserves the right to change this syllabus should it become necessary. The changes will be announced and the up-to-date syllabus will posted on this webpage.