🍩 Database of Original & Non-Theoretical Uses of Topology

1. Generalized Penalty for Circular Coordinate Representation (2020)

   Hengrui Luo, Alice Patania, Jisu Kim, Mikael Vejdemo-Johansson

   Abstract

   Topological Data Analysis (TDA) provides novel approaches that allow us to analyze the geometrical shapes and topological structures of a dataset. As one important application, TDA can be used for data visualization and dimension reduction. We follow the framework of circular coordinate representation, which allows us to perform dimension reduction and visualization for high-dimensional datasets on a torus using persistent cohomology. In this paper, we propose a method to adapt the circular coordinate framework to take into account sparsity in high-dimensional applications. We use a generalized penalty function instead of an \$L_\2\\$ penalty in the traditional circular coordinate algorithm. We provide simulation experiments and real data analysis to support our claim that circular coordinates with generalized penalty will accommodate the sparsity in high-dimensional datasets under different sampling schemes while preserving the topological structures.

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   • High-dimensional data
   • Machine learning
   • Non-linear dimension reduction
   • Sonar
   • Voting
   • Persistent cohomology

2. Geometry and Topology of the Space of Sonar Target Echos (2018)

   Michael Robinson, Sean Fennell, Brian DiZio, Jennifer Dumiak

   Abstract

   Successful synthetic aperture sonar target classification depends on the “shape” of the scatterers within a target signature. This article presents a workflow that computes a target-to-target distance from persistence diagrams, since the “shape” of a signature informs its persistence diagram in a structure-preserving way. The target-to-target distances derived from persistence diagrams compare favorably against those derived from spectral features and have the advantage of being substantially more compact. While spectral features produce clusters associated to each target type that are reasonably dense and well formed, the clusters are not well-separated from one another. In rather dramatic contrast, a distance derived from persistence diagrams results in highly separated clusters at the expense of some misclassification of outliers.

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   • Signal Processing
   • Sonar
   • Sonar echos
   • neural networks
   • Persistence diagrams
   • Persistent homology
   • Spectral features