Masterproef T713 : Computing the structured matrix geometric mean

Begeleiding:

Informatie:	Raf Vandebril
Promotor:	Raf Vandebril
Begeleider:	Ben Jeuris

Onderzoeksgroep:

Numerieke Approximatie en Lineaire Algebra Groep

Context:

The geometric mean of some numbers is defined as the n‐th root of the product of these numbers. Whereas the arithmetic mean averages some numbers, the geometric mean is more a easure to average ratios between numbers. The geometric mean is therefore a more natural mean for a whole variety of applications.

Unfortunately due to the noncommutativity of matrices, it is not straightforward to generalize this to matrices, however, in some domains such as radar technology, machine learning applied o classification of genes, ... extending the geometric mean to matrices seems the natural thing to do.

Several techniques exist, for computing the matrix geometric mean, thereby exploiting the fact that matrices lie on a manifold, the differentiability enables the application of classical optimization techniques. These techniques are well‐developed for the standard matrix case. An extra difficulty arises, however, when one wants to preserve properties of matrices. For example consider some Toeplitz matrices, when computing the mean one wants the mean to be Toeplitz as well. Otherwise the corresponding result has no physical meaning anymore.

Doel:

The aim is to study manifold optimization techniques in correspondence to computing thmatrix geometric mean. Based on the existing techniques the structure requested by several applications will be investigated and we will investigate if simple adaptations of the standard method can be made to suit the needs of the application.

Uitwerking:

The student has to study the problem of computing the matrix geometric mean. He has to learn about matrix manifold optimization and the current manifold approaches for computing geometric means. Finally the student has to investigate for which applications matrix properties can be translated to submanifolds and investigate possibilities of adapting the current existing solvers to the applications.

Profiel:

The student needs to be familiar with numerical linear algebra and needs to be able to develop software in e.g. Matlab, C++ or Fortran.

Deze masterproef is voor 1 student.