A Stable Multi-Scale Kernel for Topological Machine Learning

2014/12/21 by Jan Reininghaus, Stefan Huber, Reininghaus, Jan +5 · 17 citations
Computer Science · Mathematics · #Algebraic Topology (math.AT) #Computer Vision and Pattern Recognition (cs.CV) #FOS: Computer and information sciences #FOS: Mathematics #Machine Learning (cs.LG) #Machine Learning (stat.ML) #cs.CV #cs.LG #math.AT #stat.ML

paper · pdf · doi:10.48550/arxiv.1412.6821

arxiv created 2014/12/21 · arxiv updated 2014/12/24

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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