PersLay: A Simple and Versatile Neural Network Layer for Persistence Diagrams

Persistence diagrams, a key descriptor from Topological Data Analysis, encode and summarize all sorts of topological features and have already proved pivotal in many different applications of data science. But persistence diagrams are weakly structured and therefore constitute a difficult input for most Machine Learning techniques. To address this concern several vectorization methods have been put forward that embed persistence diagrams into either finite-dimensional Euclidean spaces or implicit Hilbert spaces with kernels. But finite-dimensional embeddings are prone to miss a lot of information about persistence diagrams, while kernel methods require the full computation of the kernel matrix. We introduce PersLay: a simple, highly modular layer of learning architecture for persistence diagrams that allows to exploit the full capacities of neural networks on topological information from any dataset. This layer encompasses most of the vectorization methods of the literature. We illustrate its strengths on challenging classification problems on dynamical systems orbit or real-life graph data, with results improving or comparable to the state-of-the-art. In order to exploit topological information from graph data, we show how graph structures can be encoded in the so-called extended persistence diagrams computed with the heat kernel signatures of the graphs.