A Stable Multi-Scale Kernel for Topological Machine Learning

Topological data analysis offers a rich source of valu-able 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 de-signing a multi-scale kernel for persistence diagrams, a sta-ble 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. Ex-periments on two benchmark datasets for 3D shape clas-sification/retrieval and texture recognition show consider-able performance gains of the proposed method compared to an alternative approach that is based on the recently in-troduced persistence landscapes. 1