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Metric graphs are special types of metric spaces used to model and represent simple, ubiquitous, geometric relations in data such as biological networks, social networks, and road networks. We are interested in giving a qualitative description of metric graphs using topological summaries. In particular, we provide a complete characterization of the 1-dimensional intrinsicČech persistence diagrams for finite metric graphs using persistent homology. Together with complementary results by Adamaszek et al., which imply results on intrinsicČech persistence diagrams in all dimensions for a single cycle, our results constitute important steps toward characterizing intrinsicČech persistence diagrams for arbitrary finite metric graphs across all dimensions.

We propose a flexible and multi-scale method for organizing, visualizing, and understanding datasets sampled from or near stratified spaces. The first part of the algorithm produces a cover tree using adaptive thresholds based on a combination of multiscale local principal component analysis and topological data analysis. The resulting cover tree nodes consist of points within or near the same stratum of the stratified space. They are then connected to form a scaffolding graph, which is then simplified and collapsed down into a spine graph. From this latter graph the stratified structure becomes apparent. We demonstrate our technique on several synthetic point cloud examples and we use it to understand song structure in musical audio data. IntroductionWe consider point cloud data, modeled as a set of points X = {x 1 , . . . , x n } in a Euclidean space R D . Such clouds are hard to analyze directly when n and/or D is large. Subsampling techniques are often used in the former situation, and dimension reduction in the latter. In both cases, much care has to be taken to ensure that the reduction in the number of points and number of dimensions does not destroy essential features of the data. While theorems exist in both contexts, they tend to make assumptions about parameters (like intrinsic dimension) that may be unknown or that may vary widely across X. The latter problem often occurs when X is not sampled from a manifold, but rather from a stratified space.A stratified space is a topological space that can be decomposed into manifold pieces (called strata), of possibly different dimension, all of which fit together in some uniform fashion. A key distinction is between maximal and non-maximal (also called singular) strata: briefly, a non-maximal stratum occurs where two or more maximal strata meet. See Figure 2 for an example. This paper proposes a novel, fast, and flexible technique for organizing, visualizing, analyzing, and understanding a point cloud that has been sampled from or near a stratified space. The technique uses a data structure called the cover tree [6] and employs techniques derived from multi-scale local principal component analysis (MLPCA, [18]) and topological data analysis (TDA,[12]). Our method summarizes the strata that make up the underlying stratified space, exhibits how the different pieces fit together, and reflects the local geometric and topological properties of the space.Instead of subsampling or reducing dimensions, we derive from X a multi-scale set of graphs, called the scaffoldings and the spine of X. At each fixed scale, a point in X belongs to a unique node in these graphs. A subset of points belongs to a common node only if the local geometry at each of those points is similar enough; that is, only if they belong to a common stratum. We also determine whether a node corresponds to a maximal or non-maximal stratum. In the former case, this suggests that a single dimension reduction technique might be employed on the points in that node. The edges of these graphs give informatio...

We consider a generic configuration of regions, consisting of a collection of distinct compact regions {Ω i } in R n+1 which may be either regions with smooth boundaries disjoint from the others or regions which meet on their piecewise smooth boundaries B i in a generic way. We introduce a skeletal linking structure for the collection of regions which simultaneously captures the regions' individual shapes and geometric properties as well as the "positional geometry" of the collection. The linking structure extends in a minimal way the individual "skeletal structures" on each of the regions. This allows us to significantly extend the mathematical methods introduced for single regions to the configuration of regions.We prove for a generic configuration of regions the existence of a special type of Blum linking structure which builds upon the Blum medial axes of the individual regions. As part of this, we introduce the "spherical axis", which is the analogue of the medial axis but for directions. These results require proving several transversality theorems for certain associated "multi-distance" and "height-distance" functions for such configurations. We show that by relaxing the conditions on the Blum linking structures we obtain the more general class of skeletal linking structures which still capture the geometric properties.The skeletal linking structure is used to analyze the "positional geometry" of the configuration. This involves using the "linking flow" to identify neighborhoods of the configuration regions which capture their positional relations. As well as yielding geometric invariants which capture the shapes and geometry of individual regions, these structures are used to define invariants which measure positional properties of the configuration such as: measures of relative closeness of neighboring regions and relative significance of the individual regions for the configuration.All of these invariants are computed by formulas involving "skeletal linking integrals" on the internal skeletal structures of the regions. These invariants are then used to construct a "tiered linking graph", which for given thresholds of closeness and/or significance, identifies subconfigurations and provides a hierarchical ordering in terms of order of significance.1991 Mathematics Subject Classification. Primary: 53A07, 58A35, Secondary: 68U05.

ABSTRACT. We introduce a method called multi-scale local shape analysis for extracting features that describe the local structure of points within a dataset. The method uses both geometric and topological features at multiple levels of granularity to capture diverse types of local information for subsequent machine learning algorithms operating on the dataset. Using synthetic and real dataset examples, we demonstrate significant performance improvement of classification algorithms constructed for these datasets with correspondingly augmented features.