טכניון מכון טכנולוגי לישראל
הטכניון מכון טכנולוגי לישראל - בית הספר ללימודי מוסמכים  
M.Sc Thesis
M.Sc StudentHadad Alon
SubjectNonnegative Matrix Factorizations
DepartmentDepartment of Applied Mathematics
Supervisor Professor Emeritus Abraham Berman


Abstract

The nonnegative matrix factorization (NMF) problem is : Given an  nonnegative matrix A, find an  nonnegative matrix U and an  nonnegative matrix V such that r is small and UV approximates A.

NMF has recently been shown to be a very useful technique in approximating high dimensional data, where the data are comprised of nonnegative component.

Because of the nonnegativity constraint, NMF has the advantage that the matrices U and V have special meaning. In the thesis, this special meaning is explained and demonstrated in the context of text mining.

There are several algorithms for NMF, in the thesis we compare the main algorithm of Lee and seug with another algorithm of  Patrick Hoyer and described the advantage of each method.

We also discuss the nonnegative rank problem : given an  A, the problem is to find the smallest integer k, such that there are tow nonnegative matrices U of order  and V of order  such that , and the completely positive rank problem : given a square nonnegative matrix A of order n find, if possible, the smallest integer k, such that  where B is a nonnegative  matrix