CS 530: Geometric and Probabilistic Methods in Computer Science *

Instructor: Lance Williams <williams@cs.unm.edu>
Time: MWF 11:00 - 11:50 AM
Location: Woodward 149
Office Hours: Mon. 4:00-5:00, Wed. 4:00-5:00.
Office: FEC 349C

Description

This is a course in applied mathematics for computer scientists, with an emphasis on information theory and linear systems theory. The goal of this course is to introduce computer science graduate students to the practical kind of mathematics useful for simulation and modeling and by researchers in scientific computation, computer vision and graphics, image processing, robotics, machine learning, and neural networks.

Course Syllabus **

Aug. 23-27
- Introduction
- Mathematical Prerequisites Exam
- Discrete Random Variables
Aug. 30-Sept. 3
Sept. 6-10
Sept. 13-17
Sept. 20-24
- Spectral Theorem (Ex. 2)
- Markov Chains (Ex. 3) (Ex. 4)
- (contd.)
Sept. 27-Oct. 1
- Information Sources with Memory
- Markov Random Fields (Ex. 5)
- Review for Midterm I
Oct. 4-8
- Midterm I
- Complex Numbers
- Fall Break
Oct. 11-15
Oct 18-22
Oct. 25-29
- Fourier Transform Theorems
- Shannon Sampling Theorem
- (contd.)
Nov. 1-5
Nov. 8-12
- Review for Midterm II
- Midterm II
- Numerical Solution of PDE's
Nov. 15-19
- Principal Component Analysis
- JPEG
- Thanksgiving
Nov. 22-26
- Linear Algebra and Color
- Frames vs. Bases
- (contd.)
Nov. 29-Dec. 3
- Intro. to Wavelet Transforms (Ex. 6)
- (contd.)
- Fast Wavelet Transform
Dec. 6-10
- (contd.)
- Discrete Orthogonal Wavelets
- (contd.)

Prerequisites

This is not a linear algebra course. Knowledge of basic linear algebra is a prerequisite! Concepts you should understand are: vector sum and difference, inner product, matrix product, matrix transpose, matrix inverse, linear independence, span, basis, rank, orthogonality, change of basis, eigenvectors, and eigenvalues. There will be an examination during the second meeting of the class on the mathematics prerequisites. Students who do poorly on the prerequisite examination should take Math 321.

Homeworks

There will be approximately six homework assignments. Many of the homework problems will be similar to those you will find on the midterms and final exams. Other problems will require experimentation in MATLAB. All are designed to increase your understanding of the fundamental ideas. Homeworks are to be turned in during class on the day they are due. They should not be emailed to the professor.

Additional Resources

Schaum's Outline of Probability, Random Variables, and Random Processes (Hsu).
Schaum's Outline of Linear Algebra (Lipschutz-Lipson).
ven der Lubbe, J.C., Information Theory, Cambridge Univ. Press, Cambridge, UK, 1997.
Reza, F.M., Introduction to Information Theory, Dover Publications, New York, NY, 1994.
Lay, D.C., Linear Algebra and Its Applications (2nd Ed.), Addison-Wesley, 2000.
Strang, G., Linear Algebra and Its Applications (3rd Ed.), Harcourt Brace Jovanovich, San Diego, CA, 1988.
Roberts, M.J., Signals and Systems, McGraw Hill, New York, NY, 2004.
Schaum's Outline of Signals and Systems (Hsu).
Hubbard, B.B., The World According to Wavelets (2nd Ed.), A.K. Peters, Wellesley, MA, 1998.

Grading

Homeworks (approx. 6): 30%
Midterm I: 18%
Midterm II: 18%
Final Exam: 34%

Miscellany

MATLAB

Most programming will be done in MATLAB or GNU Octave. Both have excellent online documentation. Here are some useful routines:

Images

Calculators

Although experience shows them to be of little actual value, on all exams (including the prerequisites exam), you are welcome to use a basic scientific calculator, e.g., a TI-30. You will not be allowed to use graphing calculators, programmable calculators, PDA's, or laptops. A good rule of thumb is that if it cost you more than $20, you probably cannot use it. If you have doubts, see me.

* This page can be found at http://www.cs.unm.edu/~williams/cs530f04.html
** Subject to change.