  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.
