Topological invariants for data: duality, stability, and applications to inference and machine learning
File(s)
Author(s)
García-Redondo, Inés
Type
Thesis
Abstract
Topological data analysis (TDA) is a modern field at the intersection of algebraic topology, statistics, and data science, focused on extracting meaningful insights from complex data where shape and size matter. A central tool in TDA is persistent homology, which extends classical homology to study topological features across multiple scales via filtrations. The resulting algebraic object, the persistence module, is summarized through topological invariants such as persistence barcodes, which record the lifespan of features in the data. In multiparameter persistence, where filtrations are indexed by multiple parameters, the theory becomes more intricate, motivating a rich line of recent research.
This thesis investigates persistent homology in both single- and multiparameter settings, combining theoretical contributions with applications in data analysis and machine learning. The work is organized into three parts.
The first part studies dualities in persistence. While persistent cohomology is well-understood and leveraged for computational efficiency in the single-parameter case, it remains underdeveloped in the multiparameter setting. We introduce a theoretical framework for dualities in multiparameter persistence and implement a cohomology-based pipeline to match barcodes from different data sources in the single-parameter case.
The second part focuses on functional invariants, which integrate more naturally into statistical and machine learning workflows than barcodes. We prove stability results for the rank invariant and establish a functional central limit theorem for the multiparameter persistence landscape, enabling the construction of confidence bands.
The final part presents two machine learning applications. First, we use persistent homology to estimate the intrinsic dimension of neural network training trajectories and relate it to generalization. Second, we apply it to interpret model behavior under adversarial attacks on large language models.
This thesis investigates persistent homology in both single- and multiparameter settings, combining theoretical contributions with applications in data analysis and machine learning. The work is organized into three parts.
The first part studies dualities in persistence. While persistent cohomology is well-understood and leveraged for computational efficiency in the single-parameter case, it remains underdeveloped in the multiparameter setting. We introduce a theoretical framework for dualities in multiparameter persistence and implement a cohomology-based pipeline to match barcodes from different data sources in the single-parameter case.
The second part focuses on functional invariants, which integrate more naturally into statistical and machine learning workflows than barcodes. We prove stability results for the rank invariant and establish a functional central limit theorem for the multiparameter persistence landscape, enabling the construction of confidence bands.
The final part presents two machine learning applications. First, we use persistent homology to estimate the intrinsic dimension of neural network training trajectories and relate it to generalization. Second, we apply it to interpret model behavior under adversarial attacks on large language models.
Version
Open Access
Date Issued
2025-08-07
Date Awarded
01/11/2025
License URL
Advisor
Monod, Anthea
Sponsor
Engineering and Physical Sciences Research Council
Grant Number
EP/S021590/1
Publisher Department
Department of Mathematics
Publisher Institution
Imperial College London
Qualification Level
Doctoral
Qualification Name
Doctor of Philosophy (PhD)