One of the main challenges in topological data analysis is to turn computed topological features, such as barcodes, into insights about the data set under analysis.
We will show in this talk how the persistent cohomology of sparse Cech filtrations (introduced recently by D. Sheehy et. al.), in dimensions 1 and 2, can be used to construct robust representations of the data in the real and complex projective spaces. Examples will be presented in order to illustrate how projective coordinates provides a framework for topology-driven nonlinear dimensionality reduction, and geometric model generation.
This work extends results of V. de Silva, D. Morozov and M. Vejdemo-Johansson on persistent cohomology and circular coordinates.