Brief Course Description
This is a year-long seminar on the general topic of sparse approximation,
focusing on two central models of sparse approximation: combinatorial
group testing, and compressive sensing.
The topics are chosen partly to fit the (research) interests of the
instructors.
- (Fall 2011) Combinatorial group testing and applications.
The basic setting of the group testing problem is to identify a subset of
"positive" items from a huge item population using as few "tests" as possible.
The meaning of "positive", "tests" and "items" are dependent on the
application. For example, dated back to World War II when the area of group
testing started, "items" are blood samples, "positive" means
syphilis-positive, and a "test" contains a pool of blood samples which
result in a positive outcome if there is at least one sample in the pool
positive for syphylis. This basic problem paradigm has found numerous
applications in biology, cryptography, networking, signal processing, coding
theory, statistical learning theory, data streaming, etc. In this semester
we introduce group testing from a computational view point, where not only the
constructions of group testing strategies are of interest, but also the
computational efficiency of both the construction and the decoding procedures
are studied. We will also briefly introduce the probabilistic method,
algorithmic coding theory, and several direct applications of group testing.
We will also cover variations of group testing and their applications.
The techniques we cover here will also be useful in compressive sensing
which is covered in the next seminar.
- (Spring 2012) Compressive sensing is based on the idea that many
signals can be represented with only a few non-zero coefficients (under
a suitably chosen basis). These signals can be "measured" using relatively
few linear measurements and can be reconstructed from the measurement vectors
efficiently. This paradigm has found numerous applications in signal processing,
data streaming, image processing, and so forth. In this part of the seminar
we shall cover the basics of compressive sensing, from efficient measurement
matrix constructions to efficient signal reconstruction. Lowerbounds with
interesting connections to communications complexity are also covered.
Instructors
- Hung Q.
Ngo ( hungngo [at] buffalo )
- Office hours: 9-10am, Mondays and Wednesdays, 238 Bell.
- Atri Rudra
( atri [at] buffalo )
- Office hours: by appointment
Prerequisites
Basic knowledge of probability
theory. (We assume that you have studied some introductory probability
course/book before.)
Work Load
Students are expected to
participate in class, and make at least one presentation. Instructors
will assign the topic and material to be presented. No A/F grade will
be given, only S/U grades.
Some reference materials (you're not required to purchase any book):
- Hung Q. Ngo, Ely Porat, and Atri Rudra,
``Efficiently Decodable Error-Correcting List Disjunct Matrices and
Applications,''
in Proceedings of The 38th International Colloquium on Automata,
Languages and Programming (ICALP 2011), July 04 -- 08, 2011,
Zurich, Switzerland.
- Piotr Indyk, Hung Q. Ngo, and Atri Rudra,
``Efficiently Decodable Non-adaptive Group Testing,''
in Proceedings of the
20th Annual ACM-SIAM Symposium on Discrete Algorithms
(SODA 2010), Austin, Texas, Jan 17-19, 2010.
-
Ding-Zhu Du and Frank Hwang,
Combinatorial Group Testing and Its Applications (Applied Mathematics)
- Hung Q. Ngo, and Ding-Zhu Du, A Survey on Combinatorial Group Testing Algorithms with Applications to DNA Library Screening, in Discrete mathematical problems with medical applications (New Brunswick, NJ), 171--182, DIMACS Ser. Discrete Math. Theoret. Comput. Sci., 55, Amer. Math. Soc., Providence, RI, 2000.
[ pdf ]. This survey is very old, and becoming
irrelevant. A new survey will come out soon, hopefully!