High-dimensional data sets often carry meaningful low-dimensional structure. There are different ways of extracting such structural information. The classic (circa 2000, with some anticipation in the 1990s) strategy of nonlinear dimensionality reduction (NLDR) involves exploiting geometric structure (geodesics, local linear geometry, harmonic forms etc) to find a small set of useful real-valued coordinates. The classic (circa 2000, with some anticipation in the 1990s) strategy of persistent topology calculates robust topological invariants based on a parametrized modification of homology theory.
In this talk, I will describe a fusion of these two strategies, in which persistent cohomology is used to find circle-valued coordinate functions. This can be thought of as a sort of "topological dimensionality reduction".
As we shall see, it is possible to exploit the resulting coordinates to study time-series data and dynamical systems. This is joint work with Dmitry Morozov, Primoz Skraba, and Mikael Vejdemo-Johansson.