## MTH 994-001: High-Dimensional Probability with Applications

Instructor: |
Mark Iwen |

Time and Place: |
Tu Th 10:20 am -- 11:40 am in Wells Hall A306. |

E-mail: |
markiwen@math.msu.edu |

Instructor Office: |
D220 WH |

Office Hours: |
Immediately after each class, and by appointment. |

This course will begin by reviewing concentration of sums of independent random variables (think the central limit theorem together with quantifications of how quickly convergence in probability occurs). From that starting point, the class of sub-Gaussian random matrices will be introduced and their singular values studied in the finite dimensional regime. Applications in numerical linear algebra and mathematical data science will be discussed in the process. Finally, random processes will be introduced and some applications involving Gaussian width to geometric problems in mathematical data science will be surveyed. The course will largely follow (in a mostly linear way) Roman Vershynin's beautiful and award winning textbook High-Dimensional Probability: An Introduction with Applications in Data Science.

#### Course website for MTH994-001:

http://math.msu.edu/~markiwen/Teaching/MTH994/Fall2022/MTH994_F22.html

The course website has the course syllabus and book.

#### Textbook:

**High-Dimensional Probability: An Introduction with Applications in Data Science**, by Roman Vershynin. Here is the version we will use.

The book is free! Download it and prosper...

#### Zoom Lecture Dates:

I will be traveling on October 20, as well as from Nov 22 -- Dec 11. These 6 courses will be held at the usual time by zoom using a link that will be emailed out to the class before those dates.

#### Homework/Projects:

Exercises will be assigned in class most days. They will *not* be collected or checked other than as part of the final oral exam for the class (see below) if the student has not chosen to do a project.

Alternatively, students may elect to work on projects (approved in advance by the instructor) instead of completing the homework assignments. A written report on the project will then be required before the end of the semester which will be defended as part of the final oral exam. The written project report will count for all the homework assigned while the student was completing their optional and instructor approved research project.

Students are encouraged to work with their peers on homework assignments. Math is a collaborative discipline and two or three minds are often better than one. However, your final homework solutions must be written up individually in your own words and then kept to help you during the final oral exam. The solutions will be useless to you if you don't understand how they work!

**Assigned Homework**

Homework from Chapters 1 and 2

Homework from Chapter 5 + Fast JL Discussion

#### Final Oral Exam:

Students must either meet with the instructor in person or via zoom. The instructor will then ask the student to solve several HW problems that were assigned in class, or else to discuss their project (if they chose to do one). The student's HW grade will be based on their participation in the exam, together with the quality of their solutions. The oral exam will be open notes -- it is expected that the student will consult their project report and/or personal homework solutions as part of the exam. In short, the purpose of this exam is to convince the instructor that the student completed ``most'' of the assigned homework (if they did not do a project), or else to describe their project, and summarize their project's conclusions, as well as to answer instructor questions about their project.

The time and place (or link) for the final oral exam will be scheduled with the instructor individually, or in small groups. All exams should take place sometime between Dec. 12 and Dec. 16, 2022, and should be scheduled before Thanksgiving (Nov. 24). Each oral exam will be 30 minutes in duration for each student. The instructor reserves the right to waive this oral exam requirement, or to shorten its duration, for any student as necessary for scheduling purposes.

#### Class Participation:

Please ask questions, answer questions, make constructive comments, and attend class. In particular, participation in class requires that you regularly attend class.

#### Grading:

Your final grade will be assigned based on the submitted homework (and/or project report), as well as on your participation (i.e., lecture attendance).

- Homework (and/or final project) grade from the Final Oral Exam -- 75%
- Class Participation -- 25%

Some of the homework exercises will be challenging -- it is only expected that the submitted homework demonstrates effort. If the student prefers, they can submit their HW assignments at the end of the course in lieu of the oral exam (or to make up for an unsatisfactory oral exam).

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