15 research outputs found
Path homologies of deep feedforward networks
We provide a characterization of two types of directed homology for
fully-connected, feedforward neural network architectures. These exact
characterizations of the directed homology structure of a neural network
architecture are the first of their kind. We show that the directed flag
homology of deep networks reduces to computing the simplicial homology of the
underlying undirected graph, which is explicitly given by Euler characteristic
computations. We also show that the path homology of these networks is
non-trivial in higher dimensions and depends on the number and size of the
layers within the network. These results provide a foundation for investigating
homological differences between neural network architectures and their realized
structure as implied by their parameters.Comment: To appear in the proceedings of IEEE ICMLA 201
Topological Graph Neural Networks
Graph neural networks (GNNs) are a powerful architecture for tackling graph
learning tasks, yet have been shown to be oblivious to eminent substructures,
such as cycles. We present TOGL, a novel layer that incorporates global
topological information of a graph using persistent homology. TOGL can be
easily integrated into any type of GNN and is strictly more expressive in terms
of the Weisfeiler--Lehman test of isomorphism. Augmenting GNNs with our layer
leads to beneficial predictive performance for graph and node classification
tasks, both on synthetic data sets, which can be classified by humans using
their topology but not by ordinary GNNs, and on real-world data