Learning metrics for persistence-based summaries and applications for graph classification

TL;DR

This paper introduces a novel weighted kernel and an optimization framework to learn optimal metrics for persistence diagrams, enhancing graph classification performance using topological data analysis.

Contribution

It develops a new weighted kernel (WKPI) and an optimization method to learn metrics for persistence summaries tailored to specific data types.

Findings

The learned kernel improves graph classification accuracy.

The framework adapts weights based on data, outperforming preset methods.

Results are competitive with or better than existing graph classification techniques.

Abstract

Recently a new feature representation and data analysis methodology based on a topological tool called persistent homology (and its corresponding persistence diagram summary) has started to attract momentum. A series of methods have been developed to map a persistence diagram to a vector representation so as to facilitate the downstream use of machine learning tools, and in these approaches, the importance (weight) of different persistence features are often preset. However often in practice, the choice of the weight function should depend on the nature of the specific type of data one considers, and it is thus highly desirable to learn a best weight function (and thus metric for persistence diagrams) from labelled data. We study this problem and develop a new weighted kernel, called WKPI, for persistence summaries, as well as an optimization framework to learn a good metric for persistence summaries. Both our kernel and optimization problem have nice properties. We further apply the learned kernel to the challenging task of graph classification, and show that our WKPI-based classification framework obtains similar or (sometimes significantly) better results than the best results from a range of previous graph classification frameworks on a collection of benchmark datasets.