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 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.