🍩 Database of Original & Non-Theoretical Uses of Topology

1. Analyzing Collective Motion With Machine Learning and Topology (2019)

   Dhananjay Bhaskar, Angelika Manhart, Jesse Milzman, John T. Nardini, Kathleen M. Storey, Chad M. Topaz, Lori Ziegelmeier

   Abstract

   We use topological data analysis and machine learning to study a seminal model of collective motion in biology [M. R. D’Orsogna et al., Phys. Rev. Lett. 96, 104302 (2006)]. This model describes agents interacting nonlinearly via attractive-repulsive social forces and gives rise to collective behaviors such as flocking and milling. To classify the emergent collective motion in a large library of numerical simulations and to recover model parameters from the simulation data, we apply machine learning techniques to two different types of input. First, we input time series of order parameters traditionally used in studies of collective motion. Second, we input measures based on topology that summarize the time-varying persistent homology of simulation data over multiple scales. This topological approach does not require prior knowledge of the expected patterns. For both unsupervised and supervised machine learning methods, the topological approach outperforms the one that is based on traditional order parameters.

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   • Biology
   • Biology:emergence
   • Biology:emergence:collective behavior
   • Collective motion
   • Order parameters
   • Time series
   • Machine learning
   • Persistent homology
   • Time series

2. Genomics Data Analysis via Spectral Shape and Topology (2022)

   Erik J. Amézquita, Farzana Nasrin, Kathleen M. Storey, Masato Yoshizawa

   Abstract

   Mapper, a topological algorithm, is frequently used as an exploratory tool to build a graphical representation of data. This representation can help to gain a better understanding of the intrinsic shape of high-dimensional genomic data and to retain information that may be lost using standard dimension-reduction algorithms. We propose a novel workflow to process and analyze RNA-seq data from tumor and healthy subjects integrating Mapper and differential gene expression. Precisely, we show that a Gaussian mixture approximation method can be used to produce graphical structures that successfully separate tumor and healthy subjects, and produce two subgroups of tumor subjects. A further analysis using DESeq2, a popular tool for the detection of differentially expressed genes, shows that these two subgroups of tumor cells bear two distinct gene regulations, suggesting two discrete paths for forming lung cancer, which could not be highlighted by other popular clustering methods, including t-SNE. Although Mapper shows promise in analyzing high-dimensional data, building tools to statistically analyze Mapper graphical structures is limited in the existing literature. In this paper, we develop a scoring method using heat kernel signatures that provides an empirical setting for statistical inferences such as hypothesis testing, sensitivity analysis, and correlation analysis.

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   • Lung cancer
   • Medicine
   • RNA-seq
   • Mapper