Date of Award
2026-05-01
Degree Name
Master of Science
Department
Mathematical Sciences
Advisor(s)
Nilotpal P. Sanyal
Abstract
This thesis introduces a novel geometry-aware framework for joint precision matrix estimation in spatially structured data. Unlike existing joint Gaussian graphical model estimators that rely on Euclidean coupling mechanisms such as fused or group graphical lasso, our approach incorporates intrinsic Riemannian geometry through geodesic distance penalties on the symmetric positive definite manifold.
The method measures similarity between precision matrices of adjacent spatial regions using the affine-invariant Riemannian distance, capturing deformations of concentration ellipsoids rather than entrywise differences. This is particularly suited for spatial transcriptomics, where regions may share dependence structures despite differences in scale or composition.
An efficient Alternating Direction Method of Multipliers algorithm is developed to handle the non-separable manifold coupling, enabling parallel updates while preserving positive definiteness. Simulation studies demonstrate that the proposed method yields stable and coherent network estimates under spatially varying yet structurally related precision matrices. Finally, the method is illustrated on a real spatial transcriptomics dataset. Overall, the framework provides a principled geometric approach for robust and interpretable precision matrix estimation in spatially structured high-dimensional data.
Language
en
Provenance
Received from ProQuest
Copyright Date
2026-05
File Size
89 p.
File Format
application/pdf
Rights Holder
Abdul Rahman Adam
Recommended Citation
Adam, Abdul Rahman, "Riemannian Manifold-Coupled Joint Sparse Graphical Models For Spatially Structured Precision Matrix Estimation" (2026). Open Access Theses & Dissertations. 4614.
https://scholarworks.utep.edu/open_etd/4614