A Stable Multi-Scale Kernel for Topological Machine Learning

About

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.

Jan Reininghaus, Stefan Huber, Ulrich Bauer, Roland Kwitt · 2014

Related benchmarks

Task	Dataset	Result
Graph Classification	MUTAG (10-fold cross-validation)	Accuracy66.5	236
Graph Classification	NCI1 (10-fold cross-validation)	Accuracy61.2	119
Graph Classification	ENZYMES (10-fold cross-validation)	Accuracy21.2	94
Classification	Airplane	Accuracy65.4	47
Classification	Texture	Accuracy98.8	33
Binary Classification	Synthesized persistence diagrams 100 (test)	Accuracy53.6	32
Graph Classification	PTC MR (10-fold cross val)	Accuracy57.6	30
Graph Classification	DHFR 10-fold CV	Accuracy61	13
Graph Classification	COX2 (10-fold CV)	Accuracy78.2	13
Topological Difference Detection	ABIDE H0 I (children < 13 vs. adults ≥ 18)	P-value0.011	12