Estimating the Topology of Low-Dimensional Data Using Deep Neural Networks

这个项目将扩展深度神经网络的超人类视觉能力，使我们能够分析三维和四维流形的拓扑结构。虽然这些空间仍然被认为是低维的，但四维流形通常超出了人类的视觉理解。流形的拓扑结构描述了它的基本属性，如连接部件、孔、隧道和各种尺寸的空腔的数量。传统的计算拓扑学方法由于计算量的限制而无法在大规模的实际应用中应用。我们提出了一种近似方法，克服了以前的限制，可以为材料科学，医学成像，动力系统和其他应用中的数据分析打开新的大门。

This project will expand on the superhuman visual capabilities of deep neural networks to allow us to analyse the topology of 3- and 4-dimensional manifolds. While these spaces still count as low-dimensional, 4-dimensional manifolds typically are beyond human visual comprehension. The topology of a manifold describes its essential properties such as the number of connected components, holes, tunnels and cavities of various dimensions. Traditional methods from computational topology fail in large practical applications due to computational restrictions. We propose an approximation that overcomes previous limitations and can open new doors to data analysis in material science, medical imaging, dynamical systems and other applications.