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Recently, neural network architectures have been developed to accommodate when the data has the structure of a graph or, more generally, a hypergraph. While useful, graph structures can be potentially limiting. Hypergraph structures in general do not account for higher order relations between their hyperedges. Simplicial complexes offer a middle ground, with a rich theory to draw on. We develop a convolutional neural network layer on simplicial 2-complexes.
Deep Neural Networks (DNNs) have achieved im- pressive accuracy in many application domains including im- age classification. Training of DNNs is an extremely compute- intensive process and is solved using variants of the stochastic gradient descent
We show that the Connes-Consani semi-norm on singular homology with real coefficients, defined via s-modules, coincides with the ordinary $ell^1$-semi-norm on singular homology in all dimensions.
In this paper we develop further the multi-parameter model of random simplicial complexes. Firstly, we give an intrinsic characterisation of the multi-parameter probability measure. Secondly, we show that in multi-parameter random simplicial complexe
There are many ways to present model categories, each with a different point of view. Here wed like to treat model categories as a way to build and control resolutions. This an historical approach, as in his original and spectacular applications of m
Given a simplicial complex K with weights on its simplices and a chain on it, the Optimal Homologous Chain Problem (OHCP) is to find a chain with minimal weight that is homologous (over the integers) to the given chain. The OHCP is NP-complete, but i