A Stable Multi-Scale Kernel for Topological Machine Learning

Jan Reininghaus, Stefan Huber, Ulrich Bauer, Roland Kwitt; The IEEE Conference on Computer Vision and Pattern Recognition (CVPR), 2015, pp. 4741-4748

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.

Related Material


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[bibtex]
@InProceedings{Reininghaus_2015_CVPR,
author = {Reininghaus, Jan and Huber, Stefan and Bauer, Ulrich and Kwitt, Roland},
title = {A Stable Multi-Scale Kernel for Topological Machine Learning},
booktitle = {The IEEE Conference on Computer Vision and Pattern Recognition (CVPR)},
month = {June},
year = {2015}
}