# Measuring and Explaining the Inter-Cluster Reliability of Multidimensional Projections

arXiv: https://arxiv.org/abs/2107.07859 · HTML: https://mlsanity.com/paper/2107.07859/

## Key facts

- Cited by 20 papers in the map (data through 14 Oct 2026).
- Builds on 7 of its 77 references that are in the map.
- First posted on arXiv 16 Jul 2021.

## Builds on (most cited first, 7 of 7)

- [Fashion-MNIST: a Novel Image Dataset for Benchmarking Machine Learning Algorithms](https://mlsanity.com/paper/1708.07747/) (2017)
- [UMAP: Uniform Manifold Approximation and Projection for Dimension Reduction](https://mlsanity.com/paper/1802.03426/) (2018)
- [Topological Autoencoders](https://mlsanity.com/paper/1906.00722/) (2019)
- [Accelerated Hierarchical Density Clustering](https://mlsanity.com/paper/1705.07321/) (2017)
- [Visualizing Large-scale and High-dimensional Data](https://mlsanity.com/paper/1602.00370/) (2016)
- [t-viSNE: Interactive Assessment and Interpretation of t-SNE Projections](https://mlsanity.com/paper/2002.06910/) (2020)
- [Interpreting Distortions in Dimensionality Reduction by Superimposing Neighbourhood Graphs](https://mlsanity.com/paper/1909.12902/) (2019)

## Built on by (most cited first, 20 of 20)

- [ZADU: A Python Library for Evaluating the Reliability of Dimensionality Reduction Embeddings](https://mlsanity.com/paper/2308.00282/) (2023)
- [Uniform Manifold Approximation with Two-phase Optimization](https://mlsanity.com/paper/2205.00420/) (2022)
- [Classes are not Clusters: Improving Label-based Evaluation of Dimensionality Reduction](https://mlsanity.com/paper/2308.00278/) (2023)
- [Feature Learning for Nonlinear Dimensionality Reduction toward Maximal Extraction of Hidden Patterns](https://mlsanity.com/paper/2206.13891/) (2022)
- [CLAMS: A Cluster Ambiguity Measure for Estimating Perceptual Variability in Visual Clustering](https://mlsanity.com/paper/2308.00284/) (2023)
- [Stop Misusing t-SNE and UMAP for Visual Analytics](https://mlsanity.com/paper/2506.08725/) (2025)
- ["Normalized Stress" is Not Normalized: How to Interpret Stress Correctly](https://mlsanity.com/paper/2408.07724/) (2024)
- [Distortion-Aware Brushing for Reliable Cluster Analysis in Multidimensional Projections](https://mlsanity.com/paper/2201.06379/) (2022)
- [FedNE: Surrogate-Assisted Federated Neighbor Embedding for Dimensionality Reduction](https://mlsanity.com/paper/2409.11509/) (2024)
- [Dataset-Adaptive Dimensionality Reduction](https://mlsanity.com/paper/2507.11984/) (2025)
- [GhostUMAP2: Measuring and Analyzing (r,d)-Stability of UMAP](https://mlsanity.com/paper/2507.17174/) (2025)
- [UMATO: Bridging Local and Global Structures for Reliable Visual Analytics with Dimensionality Reduction](https://mlsanity.com/paper/2508.16227/) (2025)
- [How Scale Breaks "Normalized Stress" and KL Divergence: Rethinking Quality Metrics](https://mlsanity.com/paper/2510.08660/) (2025)
- [Navigating High-Dimensional Backstage: A Guide for Exploring Literature for the Reliable Use of Dimensionality Reduction](https://mlsanity.com/paper/2506.14820/) (2025)
- [Why Can't I See My Clusters? A Precision-Recall Approach to Dimensionality Reduction Validation](https://mlsanity.com/paper/2509.04222/) (2025)
- [Class Angular Distortion Index for Dimensionality Reduction](https://mlsanity.com/paper/2605.00637/) (2026)
- [FADEx: Feature Attribution and Distortion-based Explanation of Dimensionality Reduction](https://mlsanity.com/paper/2607.27463/) (2026)
- [Measuring Distortion in the Empty Regions of Dimensionality Reduction Scatterplots with the Gap Index](https://mlsanity.com/paper/2607.28324/) (2026)
- [Understanding Differentiable Embeddings Through Differential and Integral Geometry](https://mlsanity.com/paper/2608.06809/) (2026)
- [A Fixed-Radius Distance-Band Benchmark for Dimensionality-Reduction Fidelity](https://mlsanity.com/paper/2608.21779/) (2026)

## Abstract (arXiv)

We propose Steadiness and Cohesiveness, two novel metrics to measure the inter-cluster reliability of multidimensional projection (MDP), specifically how well the inter-cluster structures are preserved between the original high-dimensional space and the low-dimensional projection space. Measuring inter-cluster reliability is crucial as it directly affects how well inter-cluster tasks (e.g., identifying cluster relationships in the original space from a projected view) can be conducted; however, despite the importance of inter-cluster tasks, we found that previous metrics, such as Trustworthiness and Continuity, fail to measure inter-cluster reliability. Our metrics consider two aspects of the inter-cluster reliability: Steadiness measures the extent to which clusters in the projected space form clusters in the original space, and Cohesiveness measures the opposite. They extract random clusters with arbitrary shapes and positions in one space and evaluate how much the clusters are stretched or dispersed in the other space. Furthermore, our metrics can quantify pointwise distortions, allowing for the visualization of inter-cluster reliability in a projection, which we call a reliability map. Through quantitative experiments, we verify that our metrics precisely capture the distortions that harm inter-cluster reliability while previous metrics have difficulty capturing the distortions. A case study also demonstrates that our metrics and the reliability map 1) support users in selecting the proper projection techniques or hyperparameters and 2) prevent misinterpretation while performing inter-cluster tasks, thus allow an adequate identification of inter-cluster structure.