Hierarchical simplicial manifold learning

Learning global structures, i.e. topological properties, inherent in complex data is an essential yet challenging task that spans across various scientific and engineering disciplines. A fundamental approach is to extract local data representations and use them to assemble the global structure. This conjunction of local and global operations catalyzes the integration of tools from algebraic and computational topology with machine learning. In this article, we propose a hierarchical simplicial manifold learning algorithm, constituted by nested clustering and topological reduction, for constructing simplicial complexes and decoding their topological properties. We show that the learned complex possesses the same topology as the original embedding manifold from which the data were sampled. We demonstrate applicability, convergence, and computational efficiency of the algorithm on both synthetic and real-world data.