A Stable Multi-Scale Kernel for Topological Machine Learning
This work addresses a problem for researchers in machine learning and computer vision by providing a stable kernel for topological features, though it is incremental as it builds on existing persistence diagram methods.
The paper tackled the lack of a theoretically sound connection between topological data analysis and kernel-based learning techniques by designing a stable multi-scale kernel for persistence diagrams, resulting in considerable performance gains on benchmark datasets for 3D shape classification/retrieval and texture recognition compared to an alternative method.
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.