Improving Metric Dimensionality Reduction with Distributed Topology
This work addresses the challenge of enhancing dimensionality reduction for data visualization and analysis, though it appears incremental as it builds upon existing methods like Isomap.
The authors tackled the problem of improving dimensionality reduction by proposing DIPOLE, a gradient-descent method that combines metric geometry and distributed persistent homology to correct initial embeddings, resulting in outperformance over methods like UMAP, t-SNE, and Isomap on various datasets.
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.