  Dimensionality reduction techniques are powerful tools for data preprocessing
and visualization which typically come with few guarantees concerning the
topological correctness of an embedding. The interleaving distance between the
persistent homology of Vietoris-Rips filtrations can be used to identify a
scale at which topological features such as clusters or holes in an embedding
and original data set are in correspondence. We show how optimization seeking
to minimize the interleaving distance can be incorporated into dimensionality
reduction algorithms, and explicitly demonstrate its use in finding an optimal
linear projection. We demonstrate the utility of this framework to data
visualization.
