Estimating Betti Numbers Using Deep Learning

Rahul Paul, Stephan K. Chalup

Abstract: This paper proposes an efficient computational approach for estimating the topology of manifold data as it may occur in applications. For two- or three-dimensional point cloud data, the computation of Betti numbers using persistent homology tools can already be computationally very expensive. We propose an alternative approach that employs deep learning to estimate Betti numbers of manifolds approximated by point clouds. A critical aspect in this new approach is the generation of suitable synthetic training data of scalable topological complexity. Once deep neural networks are trained on this data, inference can be computationally efficient and robust to noise. The pilot results of our study for two- and three-dimensional data support the hypothesis that deep convolutional neural networks can estimate Betti numbers of simulated data that has a topological complexity beyond immediate human visual comprehension. The approach could be generalised beyond estimating the numbers of holes, cavities and tunnels in low-dimensional manifolds to counting high-dimensional holes in high-dimensional data.