The Higher-Dimensional Skeletonization Problem

Abstract

Real data is often given as a point cloud, i.e. a finite set of points with pairwise distances between them. An important problem is to detect the topological shape of data --- for example, to approximate a point cloud by a low-dimensional non-linear subspace such as an embedded graph or a simplicial complex. Classical clustering methods and principal component analysis work well when given data points split into well-separated clusters or lie near linear subspaces of a Euclidean space. Methods from topological data analysis in general metric spaces detect more complicated patterns such as holes and voids that persist for a long time in a 1-parameter family of shapes associated to a cloud. These features can be visualized in the form of a 1-dimensional homologically persistent skeleton, which optimally extends a minimal spanning tree of a point cloud to a graph with cycles. We generalize this skeleton to higher dimensions and prove its optimality among all complexes that preserve topological features of data at any scale.

References in corpus (2)

• Further applications of the Container Method
• A fast and robust algorithm to count topologically persistent holes in noisy clouds

Cited by in corpus (5)

• Networks beyond pairwise interactions: structure and dynamics
• Randomly Weighted $d-$complexes: Minimal Spanning Acycles and Persistence Diagrams
• Skeletonisation Algorithms with Theoretical Guarantees for Unorganised Point Clouds with High Levels of Noise
• The mergegram of a dendrogram and its stability
• Parallel computation of interval bases for persistence module decomposition