Extracting insights from the shape of complex data using topology

By P. Lum, Gurjeet Singh, A. Lehman, T. Ishkanov, Mikael Vejdemo-Johansson, M. Alagappan, J. Carlsson, Gunnar E. Carlsson. Scientific Reports, 2013

592 citations recorded by the source corpus.

This paper applies topological methods to study complex high dimensional data sets by extracting shapes (patterns) and obtaining insights about them. Our method combines the best features of existing standard methodologies such as principal component and cluster analyses to provide a geometric representation of complex data sets. Through this hybrid method, we often find subgroups in data sets that traditional methodologies fail to find. Our method also permits the analysis of individual data sets as well as the analysis of relationships between related data sets. We illustrate the use of our method by applying it to three very different kinds of data, namely gene expression from breast tumors, voting data from the United States House of Representatives and player performance data from the NBA, in each case finding stratifications of the data which are more refined than those produced by standard methods.

Subfields: Biophysics, Computational Theory and Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/f8d6570df42541bbcb0dffa13e11031415057c44

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Biophysics

In Computational Theory and Mathematics

In Geometry and Topology

Citations in the corpus

An Introduction to Topological Data Analysis: Fundamental and Practical Aspects for Data Scientists (2017, methodology): “In this section we illustrate TDA methods with the Python library Gudhi 6 (Maria et al., 2014) together with popular libraries as numpy (Walt et al., 2011), scikit-learn (Pedregosa et al., 2011), pandas (McKinney et al., 2010).”
Toward enhanced understanding and projections of climate extremes using physics-guided data mining techniques (2014, background): “Dimensionality reduction techniques that utilize manifold, atomic, and topological structures derived directly from physical laws (Kpotufe, 2009, 2011; Balakrishnan et al., 2013a, b; Kpotufe and Garg, 2013; Lum et al., 2013; Wang et al., 2014) at once could make the prediction problem both more comp”
Two’s company, three (or more) is a simplex (2016, background): “In real-world systems, simplicial complexes possess richly structured patterns that can be detected and characterized using recently developed computational tools from algebraic topology (Carlsson, 2009; Lum et al, 2013), just as graph theoretic tools can be used to study networks.”
Statistical Analysis and Parameter Selection for Mapper (2017, background): “See e.g. Lum et al. (2013); Nielson et al. (2015); Rucco et al. (2015) for examples of applications.”
Identification of type 2 diabetes subgroups through topological analysis of patient similarity (2015, methodology): “It produces a more detailed and succinct description of the data set than a typical scatter plots display (80).”
Stereotypic Immune System Development in Newborn Children (2018, methodology): “Topological data analysis (TDA) provides one method for such integrated analyses and visualization of heterogeneous data types (Lum et al., 2013), such as cell frequencies, protein concentrations, and clinical metadata (Lakshmikanth et al.”