Topological data analysis for high-dimensional data in biology

Thesis

Biological and medical scientists are increasingly producing rich data sets with some high-dimensional component. In this thesis we show three new approaches to study such data.

First, in the realm of multiparameter persistent homology, we provide a new algorithm for computing the rank invariant and persistence landscapes from certain minimal presentations of 3-parameter persistence modules. In particular, this algorithm is designed to work on Gröbner bases as output by an algorithm of Bender, Gäfvert, and Lesnick. We use these advances to compute multiparameter persistence landscapes of spatiotemporal trifiltrations of dynamic metric spaces as introduced by Kim to analyse a data set arising from swarm dynamics. This work marks, to our knowledge, the first application of 3-parameter persistent homology to a problem in data science.

We then turn our attention to the problem of cell-type classification in next-generation subcellular spatial transcriptomics data. We introduce a new algorithm, called Topological Automatic Cell Types (TopACT), which uses multiscale information pooled from local neighbourhoods to produce cell-type annotations with unprecedented spatial detail. We demonstrate TopACT on both synthetic and real-world data sets, where the method shows significantly increased accuracy and, in combination with 2-parameter persistent homology, provides new insights into immune cell organisation in the mouse kidney.

Finally, we show how a recent measure in ecological diversity can be repurposed to study tissue heterogeneity in single-cell transcriptomics data. We will show how this measure provides a much-needed cell-typeagnostic tool for transcriptomics analysis, and demonstrate its utility on two single-cell data sets from embryonic development studies and a spatial transcriptomics data set of the mouse hippocampus.

Authors: Benjamin, K

• DOI: 10.5287/ora-5rzrw2n04
• Type of award: DPhil
• Level of award: Doctoral
• Awarding institution: University of Oxford
• Language: English
• Keywords: single-cell transcriptomics; similarity-sensitive diversity; topological data analysis; multiparameter persistent homology; cell-type classification; spatial transcriptomics; machine learning
• Subjects: Mathematics