TY - THES
A1 - Sembach, Lena Caroline
T1 - Riemannian Optimization for Advanced Statistical Models with Clustered Data
N2 - Modern decision making in the digital age is highly driven by the massive amount of
data collected from different technologies and thus affects both individuals as well as
economic businesses. The benefit of using these data and turning them into knowledge
requires appropriate statistical models that describe the underlying observations well.
Imposing a certain parametric statistical model goes along with the need of finding
optimal parameters such that the model describes the data best. This often results in
challenging mathematical optimization problems with respect to the modelâ€™s parameters
which potentially involve covariance matrices. Positive definiteness of covariance matrices
is required for many advanced statistical models and these constraints must be imposed
for standard Euclidean nonlinear optimization methods which often results in a high
computational effort. As Riemannian optimization techniques proved efficient to handle
difficult matrix-valued geometric constraints, we consider optimization over the manifold
of positive definite matrices to estimate parameters of statistical models. The statistical
models treated in this thesis assume that the underlying data sets used for parameter
fitting have a clustering structure which results in complex optimization problems. This
motivates to use the intrinsic geometric structure of the parameter space. In this thesis,
we analyze the appropriateness of Riemannian optimization over the manifold of positive
definite matrices on two advanced statistical models. We establish important problem-
specific Riemannian characteristics of the two problems and demonstrate the importance
of exploiting the Riemannian geometry of covariance matrices based on numerical studies.
KW - Riemannsche Geometrie
KW - Optimierung
KW - Cluster Datenanalyse
KW - Statistisches Modell
Y1 - 2023
UR - https://ubt.opus.hbz-nrw.de/frontdoor/index/index/docId/1974
UR - https://nbn-resolving.org/urn:nbn:de:hbz:385-1-19744
SP - VI
EP - 169
ER -