Dimensionality Reduction Meets Network Science: Sensemaking on UMAP's kNN Graph

2026-07-09

AI Summary

This work addresses the underutilization of high-dimensional manifold structure inherent in the internal k-nearest neighbor graph generated by UMAP. It proposes the first systematic application of network science techniques to this graph, leveraging PageRank to identify representative samples, k-core decomposition to reveal dense cores and sparse peripheries, and clustering coefficients to detect highly cohesive local neighborhoods—all without requiring additional modeling. Experiments on MNIST and Fashion-MNIST demonstrate that this approach yields deep structural insights and achieves performance on par with or superior to specialized methods such as k-medoids and HDBSCAN in tasks including sample selection and density-based structure identification.

Abstract

While UMAP is widely used for exploring high-dimensional data, typical workflows focus on its lower-dimensional embedding, largely overlooking the rich k-nearest-neighbor (kNN) graph that UMAP constructs internally. This graph encodes the data manifold in its original high-dimensional space, before the distortion that UMAP's 2D projection introduces. We demonstrate the untapped potential of this internal representation, showing how standard graph algorithms applied to this graph enhance data sensemaking: (1) PageRank identifies representative data points, (2) k-core decomposition reveals dense core regions versus sparse periphery, and (3) clustering coefficient detects tight-knit neighborhoods with highly-similar data points. Through quantitative and qualitative evaluation on MNIST and Fashion MNIST, we show that these graph-based analyses are not only practical but also competitive with or complementary to purpose-built methods (e.g., k-medoids for exemplar selection, HDBSCAN for density-based clustering).

Problem

• dimensionality reduction
• k-nearest-neighbor graph
• UMAP
• network science
• data sensemaking

Innovation

• UMAP
• kNN graph
• graph algorithms
• dimensionality reduction
• network science