Real datasets often come in the form of an URLanised cloud of points . An overall shape of such data is hard to visualise when data points are given by coordinates in a high-dimensional Euclidean space. More general data can be represented by an abstract graph with weights or distances on links between data points. The skeletonisation problem for a point cloud is to find a graph or a skeleton that correctly represents the shape of a cloud. This paper compares three algorithms that solve the data skeletonisation problem for a general cloud with topological and geometric guarantees. First, the Mapper algorithm outputs a network of interlinked clusters. Second, the \alpha-Reeb algorithm discretises the classical Reeb graphand can be applied to discrete clouds at different scales \alpha. The third algorithm HoPeS produces a Homologically Persistent Skeleton without any extra parameters. HoPeS represents the 1-dimensional shape of a cloud at any scale by the Optimality Theorem. The Reconstruction Theorem gives conditions on a noisy sample of a graph when HoPeS provides a reconstructed graph with a correct homotopy type and within a small offset of the sample. The  experiments on synthetic and real data show that HoPeS outperforms the other algorithms on several topological and geometric measures .
Data Science aims to extract meaningful knowledge from URLanised data. Real datasets usually come in the form of a cloud of points with only pairwise distances. Numerous applications require to visualise an overall shape of a noisy cloud of points sampled from a non-linear object that is more complicated than a union of disjoint clusters. The skeletonisation problem in its hardest form is to find a 1-dimensional skeleton that correctly represents a shape of the cloud. This paper compares several algorithms that solve the above skeletonisation problem for any point cloud and guarantee a successful reconstruction. For example, given a highly noisy point sample of an unknown underlying graph, a reconstructed skeleton should be geometrically close and homotopy equivalent to (has the same number of independent cycles as) the underlying graph. One of these algorithm produces a Homologically Persistent Skeleton (HoPeS) for any cloud without extra parameters. This universal skeleton contains sub-graphs that provably represent the 1-dimensional shape of the cloud at any scale . Other subgraphs of HoPeS reconstruct an unknown graph from its noisy point sample with a correct homotopy type and within a small offset of the sample. The extensive experiments on synthetic and real data reveal for the first time the maximum level of noise that allows successful graph reconstructions .
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