A $2$-parameter Persistence Layer for Learning

Cheng Xin, Soham Mukherjee, Shreyas N. Samaga, Tamal K. Dey

Keywords: topological data analysis, graph representation, persistent homology, 2-parameter persistence, graph neural network

TL;DR: A differentiable topological layer based on a novel vector representation on $2$-parameter persistence modules.

Abstract: $1$-parameter persistent homology, a cornerstone in Topological Data Analysis (TDA), studies the evolution of topological features such as cycle basis hidden in data. It has found its application in strengthening the representation power of deep learning models like Graph Neural Networks (GNN). To enrich the representations of topological features, here we propose to study $2$-parameter persistence modules induced by bi-filtration functions. In order to incorporate these representations into machine learning models, we introduce a novel vectorization on $2$-parameter persistence modules called Generalized Rank Invariant Landscape {\textsc{Gril}}. We show that this vector representation is stable and differentiable with respect to underlying filtration functions and can be easily integrated into machine learning models to augment encoding topological features. We present an algorithm to compute the vectorization and its gradients. We also test our methods on synthetic graph datasets and compare the results with some popular graph neural networks.

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