🍩 Database of Original & Non-Theoretical Uses of Topology

1. Cell Complex Neural Networks (2020)

   Mustafa Hajij, Kyle Istvan, Ghada Zamzami

   Abstract

   Cell complexes are topological spaces constructed from simple blocks called cells. They generalize graphs, simplicial complexes, and polyhedral complexes that form important domains for practical applications. We propose a general, combinatorial, and unifying construction for performing neural network-type computations on cell complexes. Furthermore, we introduce inter-cellular message passing schemes, message passing schemes on cell complexes that take the topology of the underlying space into account. In particular, our method generalizes many of the most popular types of graph neural networks.

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   • Machine learning
   • Neural network
   • Cells
   • Cell complex

2. A Topological Framework for Deep Learning (2020)

   Mustafa Hajij, Kyle Istvan

   Abstract

   We utilize classical facts from topology to show that the classification problem in machine learning is always solvable under very mild conditions. Furthermore, we show that a softmax classification network acts on an input topological space by a finite sequence of topological moves to achieve the classification task. Moreover, given a training dataset, we show how topological formalism can be used to suggest the appropriate architectural choices for neural networks designed to be trained as classifiers on the data. Finally, we show how the architecture of a neural network cannot be chosen independently from the shape of the underlying data. To demonstrate these results, we provide example datasets and show how they are acted upon by neural nets from this topological perspective.

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   • machine learning
   • machine learning:neural network
   • machine learning:neural network:classification
   • Networks
   • Topological space
   • Classical topological properties
   • Separable space