How well does Persistent Homology generalize on graphs?

Kirill Brilliantov, Amauri H Souza, Vikas Garg

Primary Area: learning on graphs and other geometries & topologies

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Keywords: Generalization Bounds, Persistence Homology, Topological Data Analysis, Graph Representation Learning, Learning Theory

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TL;DR: Persistent Homology (PH) combined with neural networks is good for graph-based predictions. We use PAC-Bayesian analysis to provide guarantees for neural network-based persistence layers (PersLay) on graphs.

Abstract: Persistent Homology (PH) is one of the pillars of topological data analysis that leverages multiscale topological descriptors to extract meaningful features from data. More recently, the combination of PH and neural networks has been successfully used to tackle predictive tasks on graphs. However, the generalization capabilities of PH on graphs remain largely unexplored. We derive a PAC-Bayesian perturbation analysis to bridge this gap. Specifically, we introduce the first data-dependent generalization guarantees for neural network-based persistence layers (PersLay). Notably, PersLay consists of a general framework that subsumes various vectorization methods of persistence diagrams in the literature. We substantiate our theoretical analysis with experimental studies and provide insights about the generalization of PH on real-world graph classification benchmarks.

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Submission Number: 7697