Global and local persistent homology for the shape and classification of biological data

Thesis

Persistent homology (PH) is an algorithmic method that allows one to study shape and higher-order interactions in high-dimensional data. Over the last decade, PH has been used in a wide variety of applications, including biology. PH considers topological invariants, such as connected components, loops, and holes, and their changes across a filtration which one can imagine as observing the data through multiple scales or resolutions. The filtration determines the questions that can be answered about the data and, in many cases, needs to be developed specifically for the problem of interest. There are also many other practical challenges when applying PH such as computational complexity and interpretation of PH output. This thesis has two parts: In the first part, we showcase how PH can be applied to two types of biological data: tumour blood vessel networks and functional neuronal networks. For tumour blood vessel networks, we develop a novel filtration that spatially characterises their structural abnormality. We show that the number of vessel loops and their distribution in the networks change over time when tumours undergo treatment with vascular targeting agents and radiation therapy. In functional neuronal networks, we find that PH can provide insight into dynamical processes in motor-learning data as well as in working-memory data from healthy versus schizophrenic human subjects. We highlight what type of information we can gain by applying persistence landscapes and persistence images to analyse and interpret the output from PH. In the second part of this thesis, we develop novel methods that consider PH locally around data points. To address computational issues when applying PH to large and noisy data sets -- both traits are commonly found in biological data -- we develop a novel landmark selection technique for point clouds. In contrast to existing methods, our subsampling process is robust to outliers and is developed specifically for PH. We further introduce a novel method that can detect geometric anomalies, such as intersections or boundaries, in point cloud data sampled from intersecting surfaces. Our detection is based on the computation of PH in local annular neighbourhoods around points and is less sensitive to the size of the local neighbourhood and surface curvature than an existing method.

Authors: Stolz-Pretzer, B

• DOI: 10.5287/ora-1ar9mr1jr
• Type of award: DPhil
• Level of award: Doctoral
• Awarding institution: University of Oxford
• Language: English
• Subjects: Mathematics; Data Science; Biology