Path homologies of deep feedforward networks
This provides a foundation for investigating homological differences between neural network architectures, which is incremental for researchers in algebraic topology and neural network theory.
The paper tackled the problem of characterizing directed homology for fully-connected feedforward neural network architectures, showing that directed flag homology reduces to simplicial homology of the underlying graph and that path homology is non-trivial and depends on layer sizes.
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.