We report on a fusion of two different strands of research from the early 2000s: NLDR (nonlinear dimensionality
reduction) and PCT (point-cloud topology). The two fields meet in theproblem of finding circular coordinates
(thus "angles") for a data set. This new technology has possible applications in signal processing and
The fusion comes about by using the NLDR approach with more general coordinates.
We find circle-valued coordinates (such as angles) by use of the persistence framework for
applying algebraic topology. I will indicate how these calculations are carried out, and give some examples
of how one can exploit the resulting coordinates to empirically study time-series data and
My collaborators in this work are Mikael Vejdemo-Johansson, Dmitriy Morozov, and Primoz Skraba.