🍩 Database of Original & Non-Theoretical Uses of Topology

1. From Trees to Barcodes and Back Again: Theoretical and Statistical Perspectives (2020)

   Lida Kanari, Adélie Garin, Kathryn Hess

   Abstract

   Methods of topological data analysis have been successfully applied in a wide range of fields to provide useful summaries of the structure of complex data sets in terms of topological descriptors, such as persistence diagrams. While there are many powerful techniques for computing topological descriptors, the inverse problem, i.e., recovering the input data from topological descriptors, has proved to be challenging. In this article we study in detail the Topological Morphology Descriptor (TMD), which assigns a persistence diagram to any tree embedded in Euclidean space, and a sort of stochastic inverse to the TMD, the Topological Neuron Synthesis (TNS) algorithm, gaining both theoretical and computational insights into the relation between the two. We propose a new approach to classify barcodes using symmetric groups, which provides a concrete language to formulate our results. We investigate to what extent the TNS recovers a geometric tree from its TMD and describe the effect of different types of noise on the process of tree generation from persistence diagrams. We prove moreover that the TNS algorithm is stable with respect to specific types of noise.

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   • Neurology
   • Point cloud
   • metric space
   • Barcode
   • Persistence images
   • Topological Morphology Descriptor
   • Topological Neuron Synthesis

2. Extracting Insights From the Shape of Complex Data Using Topology (2013)

   P. Y. Lum, G. Singh, A. Lehman, T. Ishkanov, M. Vejdemo-Johansson, M. Alagappan, J. Carlsson, G. Carlsson

   Abstract

   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.

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   • Breast cancer
   • Gene expression
   • Medicine
   • Player performance
   • Politics
   • Shape recognition
   • Sport
   • Voting
   • Gene expression
   • Networks
   • Votes
   • coordinate-free data
   • metric space
   • Mapper

3. Lipschitz Functions Have Lp-Stable Persistence (2010)

   David Cohen-Steiner, Herbert Edelsbrunner, John Harer, Yuriy Mileyko

   Abstract

   We prove two stability results for Lipschitz functions on triangulable, compact metric spaces and consider applications of both to problems in systems biology. Given two functions, the first result is formulated in terms of the Wasserstein distance between their persistence diagrams and the second in terms of their total persistence.

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   • Gene expression
   • Gene expression
   • metric space
   • Lipschitz functions
   • Persistence diagrams
   • Persistent homology
   • Wasserstein distance
   • Wasserstein stability