Mathematical Toolkit - Autumn 2023

TTIC 31150/CMSC 31150

T Th 9:30 - 10:50 am (TTIC 530)

Discussion: W 4:30-5:30 pm (TTIC 529)

Office Hours: T 11 am - 12 pm (Madhur, TTIC 505)

The course is aimed at first-year graduate students and advanced undergraduates. The goal of the course is to collect and present important mathematical tools used in different areas of computer science. The course will mostly focus on linear algebra and probability. We intend to cover the following topics and examples:

• Abstract linear algebra: vector spaces, linear transformations, Hilbert spaces, inner product, Gram-Schmidt orthogonalization, eigenvalues and eigenvectors, SVD.
• Least squares, iterative solvers for systems of linear equations, spectral graph theory, perturbation and stability of eigenvalues
• Discrete probability: random variables, Markov, Chebyshev and Chernoff-Hoeffding bounds
• Gaussian variables, concentration inequalities, dimension reduction.
• Spectral partitioning and clustering.
• Additional topics (to be chosen from based on time and interest): Martingales, Markov Chains, Random Matrices, Tensors, Chaining Methods

The course will have 5 homeworks (60 percent), a midterm (15 percent), and a final (25 percent).

There is no textbook for this course. Please see the "Resources" and "Recommended Books" sections below for some useful references.

Homeworks and Announcements

Lecture Plan and Notes

• 9/26: Fields, vector spaces, span and bases. Lagrange interpolation.
[Notes]
• 9/28: Bases and dimension of vector spaces. Linear transformations.
[Notes]
• 10/3: Rank-nullity theorem. Eigenvalues and eigenvectors. Inner Products.
[Notes]
• 10/5: Orthogonality and orthonormality. Fourier coefficients.
[Notes]
[Notes]
• 10/12: Real Spectral Theorem. Rayleigh quotients.
[Notes]
• 10/17: Rayleigh quotients. Positive semidefinite operators. Intro to SVD.
[Notes]
• 10/19: Singular Value Decomposition. Low-rank approximation
[Notes]
• 10/24: Least squares approximation using SVD, Gershgorin disc theorem and applications
[Notes]
• 10/26: Solving systems of linear equations, steepest descent, conjugate gradient method
[Notes]
• 10/31: Basics of probability: the finite case
[Notes]
• 11/2: Independence, discrete random variables, coupon collection
[Notes]
• 11/7: Polynomial identity lemma, probabilistic method
[Notes]
• 11/14: Markov's and Chebyshev's inequalities. Threshold phenomena. Chernoff-Hoeffding bounds.
[Notes]
• 11/15: Applications of Chernoff/Hoeffding bounds. Probability and random variables over uncountable spaces.
[Notes]
• 11/16: Gaussian random variables.
[Notes]

Resources

Recommended Books

There is no textbook for this course. The following books may be useful as references: