A stable multi-scale kernel for topological machine learning

By Jan Reininghaus, Stefan Huber, Ulrich Bauer, Roland Kwitt. Computer Vision and Pattern Recognition, 2014

392 citations recorded by the source corpus.

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.

Subfields: Computational Theory and Mathematics, Computer Vision and Pattern Recognition, Radiology, Nuclear Medicine and Imaging

View Paper: https://www.semanticscholar.org/paper/09d479e266b76da83deb5e044034d9880fc5c211

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Citations in the corpus

A roadmap for the computation of persistent homology (2015, methodology): “atistical analysis and machine-learning techniques. Such methods include persistence landscapes [24], using the space of algebraic functions [3], persistence images [39], and kernelization techniques [20,107,138,172]. See the papers [102,148] for applications of persistence landscapes. The package P”
Wasserstein Stability for Persistence Diagrams (2020, positive): “Lipschitz stability with respect to 1-Wasserstein distance for persistence diagrams has been shown for a number of linear representations, see for example persistence scale space kernel [36] and persistence images [15].”
An Introduction to Topological Data Analysis: Fundamental and Practical Aspects for Data Scientists (2017, background): “Robinson and Turner (2017) focused on pairwise distances of persistence diagrams, whereas Berry et al.”
Topology-Preserving Deep Image Segmentation (2019, neutral): “For completeness, we also refer to other existing works on topological features and their kernels [1, 33, 23, 5].”
Persistent Homology Analysis of Brain Artery Trees (2014, methodology): “These results are stronger than those in Shen et al. (2014), the only other study to find a statistically significant sex difference in this data set.”
Deep Learning with Topological Signatures (2017, methodology): “Interestingly, the stability result of Theorem 1 is comparable to the stability results in [1] or [27] (which are also w.”