The shape of data: statistical topology across biology and AI
File(s)
Author(s)
Wang, Qiquan
Type
Thesis
Abstract
Topological data analysis (TDA) provides multiscale summaries of complex data, capturing topological features such as connected components, loops, and higher-dimensional structures beyond classical geometric or statistical summaries. Persistent homology (PH), a central tool in TDA, encodes these features across scales. Its algebraic invariants, such as persistence diagrams, are powerful but inhabit non-Euclidean spaces that complicate statistical treatment. To make PH compatible with inference, there is a pressing need for representations and models that balance the faithfulness of topological information with statistical tractability and computational efficiency.
This thesis develops statistical methodologies towards this need, spanning theory, methodology and applications. At the theoretical level, this thesis studies an alternative pre-existing invariant, the rank function, establishing new guarantees that bring them to practical use, ensuring interpretability and reliability, and thereby positioning them as functional summaries directly amenable to functional data analysis. Empirical studies show outperformance over common vectorisations while preserving complete topological information. Deriving asymptotic properties of another functional summary, the multiparameter persistence landscapes, provides a principled route to uncertainty quantification. At the methodological level, this thesis proposes a Topological Gaussian Mixture Model (TGMM), a probabilistic framework that treats persistence diagram points as weighted observations to embed topological information within an interpretable model. At the applied level, methods were validated in two domains. In biology, TGMM provided an interpretable probabilistic description of vascular remodelling. In machine learning, PH summaries uncovered consistent multiscale signatures of adversarial perturbations in large language models and enabled a systematic critique of PH-based generalisation measures.
Together, these contributions demonstrate that TDA can be placed on a firm statistical foundation and developed into practical inferential tools. By advancing functional and model-based approaches in real-world settings, this work contributes to broader research in statistical topology: a discipline uniting algebraic invariants, probability theory, and applications in the life sciences and AI.
This thesis develops statistical methodologies towards this need, spanning theory, methodology and applications. At the theoretical level, this thesis studies an alternative pre-existing invariant, the rank function, establishing new guarantees that bring them to practical use, ensuring interpretability and reliability, and thereby positioning them as functional summaries directly amenable to functional data analysis. Empirical studies show outperformance over common vectorisations while preserving complete topological information. Deriving asymptotic properties of another functional summary, the multiparameter persistence landscapes, provides a principled route to uncertainty quantification. At the methodological level, this thesis proposes a Topological Gaussian Mixture Model (TGMM), a probabilistic framework that treats persistence diagram points as weighted observations to embed topological information within an interpretable model. At the applied level, methods were validated in two domains. In biology, TGMM provided an interpretable probabilistic description of vascular remodelling. In machine learning, PH summaries uncovered consistent multiscale signatures of adversarial perturbations in large language models and enabled a systematic critique of PH-based generalisation measures.
Together, these contributions demonstrate that TDA can be placed on a firm statistical foundation and developed into practical inferential tools. By advancing functional and model-based approaches in real-world settings, this work contributes to broader research in statistical topology: a discipline uniting algebraic invariants, probability theory, and applications in the life sciences and AI.
Version
Open Access
Date Issued
2025-09-30
Date Awarded
01/01/2026
License URL
Advisor
Monod, Anthea
Williams, Matthew
Publisher Department
Department of Mathematics
Publisher Institution
Imperial College London
Qualification Level
Doctoral
Qualification Name
Doctor of Philosophy (PhD)