Local Topological Data Analysis to Uncover the Global Structure of Data Approaching Graph-Structured Topologies

  • Robin VandaeleEmail author
  • Tijl De Bie
  • Yvan Saeys
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11052)


Gene expression data of differentiating cells, galaxies distributed in space, and earthquake locations, all share a common property: they lie close to a graph-structured topology in their respective spaces [1, 4, 9, 10, 20], referred to as one-dimensional stratified spaces in mathematics. Often, the uncovering of such topologies offers great insight into these data sets. However, methods for dimensionality reduction are clearly inappropriate for this purpose, and also methods from the relatively new field of Topological Data Analysis (TDA) are inappropriate, due to noise sensitivity, computational complexity, or other limitations. In this paper we introduce a new method, termed Local TDA (LTDA), which resolves the issues of pre-existing methods by unveiling (global) graph-structured topologies in data by means of robust and computationally cheap local analyses. Our method rests on a simple graph-theoretic result that enables one to identify isolated, end-, edge- and multifurcation points in the topology underlying the data. It then uses this information to piece together a graph that is homeomorphic to the unknown one-dimensional stratified space underlying the point cloud data. We evaluate our method on a number of artificial and real-life data sets, demonstrating its superior effectiveness, robustness against noise, and scalability. Code related to this paper is available at:


Topological Data Analysis Persistent homology Metric spaces Graph theory Stratified spaces 



This work was funded by the ERC under the European Union’s Seventh Framework Programme (FP7/2007-2013) / ERC Grant Agreement no. 615517, and the FWO (G091017N, G0F9816N).


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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.IDLab, Department of Electronics and Information SystemsGhent UniversityGentBelgium
  2. 2.Data Mining and Modelling for Biomedicine (DaMBi)VIB Inflammation Research CenterGentBelgium

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