Date of Award:

8-2026

Document Type:

Dissertation

Degree Name:

Doctor of Philosophy (PhD)

Department:

Mathematics and Statistics

Committee Chair(s)

Kevin R. Moon

Committee

Kevin R. Moon

Committee

Alan Wisler

Committee

Brennan Bean

Committee

Hamid Karimi

Committee

Yan Sun

Abstract

Modern datasets often contain many measured variables for each observation, such as gene-expression levels, brain activity signals, or features in tabular data. These data are also often noisy, meaning that useful patterns are mixed with measurement error or irrelevant variation. Although such datasets can appear complex, they are frequently represented by simpler hidden structures, such as trajectories, clusters, or relationships between observations. This dissertation develops methods for uncovering these hidden structures by learning geometric and graph-based representations directly from data. The first part introduces Functional Information Geometry, which represents local patterns in high-dimensional data using functional features and constructs a robust geometry through a data-adaptive distance measure. This method helps recover meaningful structure from noisy scientific and dynamical data and is further extended to settings without natural time ordering, including single-cell transcriptomic data and contextual embedding data. The second part introduces Random-Forest-Induced Graph Neural Networks, which use Random Forest proximities to build graph representations for tabular data, enabling graph neural networks to be applied when no explicit relational structure is available. Together, these methods provide flexible tools for transforming complex data into more structured and interpretable forms for visualization, scientific inference, and machine learning.

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