🍩 Database of Original & Non-Theoretical Uses of Topology

1. Kernel Method for Persistence Diagrams via Kernel Embedding and Weight Factor (2017)

   Genki Kusano, Kenji Fukumizu, Yasuaki Hiraoka

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   • Granular packing system
   • Packing
   • Kernel
   • Kernel embedding
   • Persistence diagrams
   • Persistence images
   • Persistence landscape
   • Persistence weighted Gaussian kernel
   • Persistent homology

2. Persistence Weighted Gaussian Kernel for Topological Data Analysis (2016)

   Genki Kusano, Yasuaki Hiraoka, Kenji Fukumizu

   Abstract

   Topological data analysis (TDA) is an emerging mathematical concept for characterizing shapes in complex data. In TDA, persistence diagrams are widely recognized as a useful descriptor of data, and...

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   • Oxide glasses
   • Protein
   • Circles
   • Persistence Diagrams
   • proteins
   • Kernel
   • Persistence diagrams
   • Persistent homology

3. Coverage Criterion in Sensor Networks Stable Under Perturbation (2014)

   Yasuaki Hiraoka, Genki Kusano

   Abstract

   To the coverage problem of sensor networks, V. de Silva and R. Ghrist (2007) developed several approaches based on (persistent) homology theory. Their criteria for the coverage are formulated on the Rips complexes constructed by the sensors, in which their locations are supposed to be fixed. However, the sensors are in general affected by perturbations (e.g., natural phenomena), and hence the stability of the coverage criteria should be also discussed. In this paper, we present a coverage theorem stable under perturbation. Furthermore, we also introduce a method of eliminating redundant cover after perturbation. The coverage theorem is derived by extending the Rips interleaving theorem studied by F. Chazal, V. de Silva, and S. Oudot (2013) into an appropriate relative version.

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   • Coverage
   • Sensor networks
   • Persistent homology