Learning to See Topological Properties in 4D Using Convolutional Neural Networks

Topology describes the essential structure of a space, and in 4D, a larger variety of topologically distinct manifolds can be embedded versus 2D or 3D. The present study investigates an end-to-end visual approach, which couples data generation software and convolutional neural networks (CNNs) to estimate the topology of 4D data. A synthetic 4D training data set is generated with the use of several manifolds, and then labelled with their associated Betti numbers by using techniques from algebraic topology. Several approaches to implementing a 4D convolution layer are compared. Experiments demonstrate that already a basic CNN can be trained to provide estimates for the Betti numbers associated with the number of one-, two-, and three-dimensional holes in the data. Some of the intricacies of topological data analysis in the 4D setting are also put on view, including aspects of persistent homology.