Abstract Persistent homology of a point cloud is often computed to recover the topology of an underlying continuous geometric object. But this is only possible when the connectivity radius of the point cloud stays small. Could there be more to persistent homology, when the connectivity radius grows large? In this paper I bound probabilities that a random Čech complex built on a circle attains high-dimensional topology. This builds on the known result that any nerve complex of circular arcs has the homotopy type of a bouquet of spheres. We observe a phase transition going from one 1-sphere, bouquet of 2-spheres, one 3-sphere, bouquet of 4-spheres, and so on. Furthermore, the even-dimensional Betti numbers become arbitrarily large over shrinking intervals. Our main tool is an exact computation of the expected Euler characteristic, combined with constraints on homotopy types.

Classical unsupervised learning methods like clustering and linear dimensionality reduction parametrize large-scale geometry when it is discrete or linear, while more modern methods from manifold learning find low dimensional representation or infer local geometry by constructing a graph on the input data. More recently, topological data analysis popularized the use of simplicial complexes to represent data topology with two main methodologies: topological inference with geometric complexes and large-scale topology representation with Mapper graphs -- central to these is the nerve construction from topology, which builds a simplicial complex given any cover of a space by subsets. While successful, these have limitations: geometric complexes scale poorly with data size, and Mapper graphs can be hard to tune and only contain low dimensional information. In this paper, we propose to study the problem of learning covers in its own right, and from the perspective of optimization. We describe a method to learn topologically-faithful covers of geometric datasets, and show that the simplicial complexes thus obtained can outperform standard topological inference approaches in terms of size, and Mapper-type algorithms in terms of representation of large-scale topology.

A paradigm in topological data analysis asserts that persistent homology should be computed to recover the homology of a data manifold. But could there be more to persistent homology? In this paper I bound probabilities that a random m Čech complex built on a circle attains high-dimensional topology. This builds on the known result that any nerve complex of circular arcs has the homotopy type of a bouquet of spheres. We observe a phase transition going from one 1-sphere, bouquet of 2-spheres, one 3-sphere, bouquet of 4-spheres, and so on. Furthermore, the even-dimensional Betti numbers become arbitrarily large over shrinking intervals. Our main tool is an exact computation of the expected Euler characteristic, combined with constraints on homotopy types. The systematic behaviour we observe cannot be regarded as a "topological noise", and calls for deeper investigations from the TDA community.