Dimensionality reduction for homological stability and global structure preservation

Abstract

We propose DiRe, a force-directed dimensionality reduction framework designed to preserve global structure and homological features while remaining practical on modern hardware. The method combines an initial embedding with a graph-based layout optimization and evaluates the resulting low-dimensional representation using local distortion, context preservation, and persistent homology measures. Across the benchmark suite considered here, DiRe provides a complementary tradeoff to UMAP and tSNE: it is designed less as a purely local visualization heuristic and more as a framework for embeddings whose large-scale geometry can be quantified through Betti curves and persistence diagrams.

33 pages, 14 figures, 5 tables; Github repository available at https://github.com/sashakolpakov/dire-jax Reproducibility suite https://github.com/sashakolpakov/homological-stability-repro Package available on PyPi https://pypi.org/project/dire-jax/