  We propose a novel approach to dimensionality reduction combining techniques
of metric geometry and distributed persistent homology, in the form of a
gradient-descent based method called DIPOLE. DIPOLE is a
dimensionality-reduction post-processing step that corrects an initial
embedding by minimizing a loss functional with both a local, metric term and a
global, topological term. By fixing an initial embedding method (we use
Isomap), DIPOLE can also be viewed as a full dimensionality-reduction pipeline.
This framework is based on the strong theoretical and computational properties
of distributed persistent homology and comes with the guarantee of almost sure
convergence. We observe that DIPOLE outperforms popular methods like UMAP,
t-SNE, and Isomap on a number of popular datasets, both visually and in terms
of precise quantitative metrics.
