Course overview: Introduction to computational mathematics covering topics in (numerical) linear algebra, statistics, and optimization. Topics included are those of particular relevance to upper-division computer science courses in machine learning, numerical analysis, graphics, vision, robotics, and more. An emphasis is placed both on understanding core mathematical concepts and introducing associated computational methodologies.
Instructor: Anil Damle
Office hours: Mondays from 2:45 to 3:45 pm and Wednesdays from 2 to 3 pm in Gates 423
TA: Ian Delbridge
Office hours: Tuesdays 10 to 11 am in Rhodes 412 and Thursdays 12 to 1 pm in Rhodes 406
TA: Anmol Kabra
Office hours: Mondays 10 to 11 am in Rhodes 412
Lectures: Monday, Wednesday, and Friday from 11:15 AM till 12:05 PM in Gates Hall G01
Your grade in this course is comprised of three components: homeworks, exams, and a final project. Please also read through the given references in concert with the lectures.
There will be a number of homework assignments throughout the course, typically made available roughly one to two weeks before the due date. They will include a mix of mathematical questions and basic implementation of algorithms. Any required implementation may be done in MATLAB, Julia, or Python. Homeworks should be typeset (or, for the theoretical questions, very neatly handwritten and scanned) and submitted along with any associated code via the CMS.
The goal of this course is to provide you with the mathematical and computational fundamentals that underpin a broad range of applications. Therefore, in lieu of a final exam this course will have an open ended final project. In particular, leveraging what you have learned about linear algebra, statistics, and optimization you will tackle a problem of interest to you (you may work individually or in a group and the maximum group size is TBD). Several example projects spanning a range of applications will be provided, though you may also propose your own. More details about the project will be provided within the first several weeks of class including specific requirements and due dates for proposals, check-ins, etc. Per university policy, the final project will be due December 20, 2019 at 11:30 AM.
There will be two short prelim exams for this class focused on the more theoretical material discussed in the first two thirds of the course. The specific scope of each exam will be provided closer to the exam date along with relevant practical information.
Your final grade in the course will be computed based on the homework assignments, project, exams, and course participation
You are encouraged to actively participate in class. This can take the form of asking questions in class, responding to questions to the class, and actively asking/answering questions on the online discussion board. We will also be soliciting feedback mid-semester to hopefully improve the course.
You may discuss the homework and projects freely with other students, but please refrain from looking at code or writeups by others. You must ultimately implement your own code and write up your own solution.
Except for the final project report, all work is due at 11:59 pm on the due date. Homework and projects should be submitted via the CMS. For each assignment you are allowed up to two "slip days". However, over the course of the semester you may only use a total of eight slip days. You may not use slip days for the final project report.
Grades will be posted to the CMS, and regrade requests must be submitted within one week.
MATH 2210 or MATH 2940 or equivalent; pre- or co- requisite: one programming class, and some familiarity with probability and statistics
The Cornell Code of Academic Integrity applies to this course.
In compliance with the Cornell University policy and equal access laws, I am available to discuss appropriate academic accommodations that may be required for student with disabilities. Requests for academic accommodations are to be made during the first three weeks of the semester, except for unusual circumstances, so arrangements can be made. Students are encouraged to register with Student Disability Services to verify their eligibility for appropriate accommodations.
The text by Gilbert Strang serves as the primary reference for this course, though I will populate this section of the website with a collection of other resources you may find useful throughout the course.
A tentative schedule follows, and includes the topics we will be covering, relevant reference material, and assignment information. It is quite possible the specific topics covered on a given day will change slightly. This is particularly true for the lectures in the latter part of the course, and this schedule will be updated as necessary.
|9/4||Notation and fundamentals||GS: 1.1 and 1.2|
|9/6||Four fundamental subspaces||GS: 1.3|
|9/9||Orthogonal matrices and subspaces||GS: 1.3 and 1.5|
|9/13||The SVD||GS 1.8||HW 1 due|
|9/16||The SVD||GS 1.8||SVD notes|
|9/18||The SVD||GS 1.8||SVD demo code|
|9/20||Projection matrices and Gram-Schmidt||See additional resources and references||Orthogonal projector notes|
|9/23||Eigenvalues and eigenvectors||See additional resources and references||HW 2 due on 9/24|
|9/25||Eigenvalues and eigenvectors||See additional resources and references|
|9/27||Power metho||See additional resources and references|
|9/30||Sensitivity of linear systems||See additional resources and references||Condition number notes|
|10/2||Matrix factorizations and linear systems||See additional resources and references|
|10/4||Least Squares||See additional resources and references|
|10/9||Randomized methods||HW 3 due on 10/10|
|10/17||Prelim exam at 7:30 PM||Prelim exam at 7:30 PM|
|10/21||Intro to probability||GS 5.1-5.3|
|10/23||Intro to probability||GS 5.1-5.3|
|10/28||Limit theorems and multivariate Gaussian||GS 5.5|
|11/4||Estimation||GS 5.1-5.3||Homework 4 due|
|11/11||PCA||Intro to Statistical Learning 10||Homework 5 due|
|11/13||PCA||Intro to Statistical Learning 10|
|11/14||Prelim exam at 7:30 PM||Prelim exam at 7:30 PM|
|11/15||Least squares with noise||Intro to Mathematical Statistics 12.3|
|11/18||What it means to solve an optimization problem||Ascher and Greif 3.5 and 9.2, and GS 6.1 and 6.4|
|11/20||Search direction methods||Ascher and Greif 3.5 and 9.2, and GS 6.1 and 6.4|
|11/22||Search direction methods||Ascher and Greif 3.5 and 9.2, and GS 6.1 and 6.4|
|11/25||Optimization demo||demo code|
|12/2||No class, snow day|
|12/6||Stochastic Gradient Descent|
|12/9||Flex||Homework 6 due|
|12/20||Project due at 11:30 AM||Project due at 11:30 AM|