PersLay: A Neural Network Layer for Persistence Diagrams andNew Graph Topological Signatures

Mathieu Carrière, Frédéric Chazal, Yuichi Ike, Théo Lacombe, Martin Royer, Yuhei Umeda

Abstract: Persistence diagrams, the most common de-scriptors of Topological Data Analysis, en-code topological properties of data and havealready proved pivotal in many different ap-plications of data science. However, since themetric space of persistence diagrams is notHilbert, they end up being difficult inputs formost Machine Learning techniques. To ad-dress this concern, several vectorization meth-ods have been put forward that embed persis-tence diagrams into either finite-dimensionalEuclidean space or implicit infinite dimen-sional Hilbert space with kernels.In this work, we focus on persistence diagramsbuilt on top of graphs. Relying on extendedpersistence theory and the so-called heat ker-nel signature, we show how graphs can beencoded by (extended) persistence diagramsin a provably stable way. We then propose ageneral and versatile framework for learningvectorizations of persistence diagrams, whichencompasses most of the vectorization tech-niques used in the literature. We finally show-case the experimental strength of our setup byachieving competitive scores on classificationtasks on real-life graph datasets.