Multi-Scale Projective Coordinates via Persistent Cohomology of Sparse Filtrations

We present in this paper a framework for mapping data onto real and complex projective spaces. The resulting projective coordinates provide a multi-scale representation of the data, and capture low dimensional underlying topological features. An initial map is obtained in two steps: First, the persistent cohomology of a sparse filtration is used to compute systems of transition functions for (real and complex) line bundles over neighborhoods of the data. Next, the transition functions are used to produce explicit classifying maps for the induced bundles. A framework for dimensionality reduction in projective space (Principal Projective Components) is also developed, aimed at decreasing the target dimension of the original map. Several examples are provided as well as theorems addressing choices in the construction.