🍩 Database of Original & Non-Theoretical Uses of Topology

1. A Stable Multi-Scale Kernel for Topological Machine Learning (2015)

   Jan Reininghaus, Stefan Huber, Ulrich Bauer, Roland Kwitt

   Abstract

   Topological data analysis offers a rich source of valuable information to study vision problems. Yet, so far we lack a theoretically sound connection to popular kernel-based learning techniques, such as kernel SVMs or kernel PCA. In this work, we establish such a connection by designing a multi-scale kernel for persistence diagrams, a stable summary representation of topological features in data. We show that this kernel is positive definite and prove its stability with respect to the 1-Wasserstein distance. Experiments on two benchmark datasets for 3D shape classification/retrieval and texture recognition show considerable performance gains of the proposed method compared to an alternative approach that is based on the recently introduced persistence landscapes.

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   • Shape classification
   • Shape recognition
   • Texture recognition
   • Meshes
   • Simplicial complexes
   • Kernels
   • Persistence diagrams
   • Persistent homology