ClusterGraph: a new tool for visualisation and compression of multidimensional data

Understanding the organization of high-dimensional data is of primary interest for many branches of applied sciences. It is typically achieved by applying dimensionality reduction techniques, which, while preserving local features, often miss the global structure of the dataset. Clustering techniques are another class of methods operating in the ambient space, grouping together similar points. However, unlike dimensionality reduction techniques, they do not provide information about the organization of the data.

Leveraging ideas from Topological Data Analysis, we introduce ClusterGraph, an additional layer on the output of any clustering algorithm that represents clusters as vertices and inter-cluster relationships as weighted edges. This structure captures the large-scale organisation of a dataset without forcing it into a two-dimensional Euclidean embedding. The method comes with a built-in quality criterion, metric distortion, which quantifies how faithfully the graph reflects the intrinsic geometry of the data and provides a principled basis for pruning, model selection, and parameter tuning. ClusterGraph, possibly with an appropriate structure-preserving simplification, can be visualized and used in synergy with state-of-the-art exploratory data analysis techniques.

ClusterGraph is complementary to methods such as UMAP, t-SNE, and PHATE: by encoding inter-cluster geometry directly in graph form, it serves not only as a visualisation tool but also as a diagnostic layer that can validate, refine, or question conclusions drawn from low-dimensional embeddings.