# GhostUMAP2: Measuring and Analyzing (r,d)-Stability of UMAP

arXiv: https://arxiv.org/abs/2507.17174 · HTML: https://mlsanity.com/paper/2507.17174/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 35 references that are in the map.
- First posted on arXiv 23 Jul 2025.

## Builds on (most cited first, 13 of 13)

- [Fashion-MNIST: a Novel Image Dataset for Benchmarking Machine Learning Algorithms](https://mlsanity.com/paper/1708.07747/) (2017)
- [DistilBERT, a distilled version of BERT: smaller, faster, cheaper and lighter](https://mlsanity.com/paper/1910.01108/) (2019)
- [Distributed Representations of Words and Phrases and their Compositionality](https://mlsanity.com/paper/1310.4546/) (2013)
- [UMAP: Uniform Manifold Approximation and Projection for Dimension Reduction](https://mlsanity.com/paper/1802.03426/) (2018)
- [Character-level Convolutional Networks for Text Classification](https://mlsanity.com/paper/1509.01626/) (2015)
- [LINE: Large-scale Information Network Embedding](https://mlsanity.com/paper/1503.03578/) (2015)
- [Deep Learning for Classical Japanese Literature](https://mlsanity.com/paper/1812.01718/) (2018)
- [Visualizing Large-scale and High-dimensional Data](https://mlsanity.com/paper/1602.00370/) (2016)
- [Measuring and Explaining the Inter-Cluster Reliability of Multidimensional Projections](https://mlsanity.com/paper/2107.07859/) (2021)
- [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)
- [Feature Learning for Nonlinear Dimensionality Reduction toward Maximal Extraction of Hidden Patterns](https://mlsanity.com/paper/2206.13891/) (2022)
- [Visual Analytics of Multivariate Networks with Representation Learning and Composite Variable Construction](https://mlsanity.com/paper/2303.09590/) (2023)

## Built on by (most cited first, 2 of 2)

- [UMATO: Bridging Local and Global Structures for Reliable Visual Analytics with Dimensionality Reduction](https://mlsanity.com/paper/2508.16227/) (2025)
- [Dimensionality Reduction Considered Harmful (Some of the Time)](https://mlsanity.com/paper/2512.18230/) (2025)

## Abstract (arXiv)

Despite the widespread use of Uniform Manifold Approximation and Projection (UMAP), the impact of its stochastic optimization process on the results remains underexplored. We observed that it often produces unstable results where the projections of data points are determined mostly by chance rather than reflecting neighboring structures. To address this limitation, we introduce (r,d)-stability to UMAP: a framework that analyzes the stochastic positioning of data points in the projection space. To assess how stochastic elements, specifically initial projection positions and negative sampling, impact UMAP results, we introduce "ghosts", or duplicates of data points representing potential positional variations due to stochasticity. We define a data point's projection as (r,d)-stable if its ghosts perturbed within a circle of radius r in the initial projection remain confined within a circle of radius d for their final positions. To efficiently compute the ghost projections, we develop an adaptive dropping scheme that reduces a runtime up to 60% compared to an unoptimized baseline while maintaining approximately 90% of unstable points. We also present a visualization tool that supports the interactive exploration of the (r,d)-stability of data points. Finally, we demonstrate the effectiveness of our framework by examining the stability of projections of real-world datasets and present usage guidelines for the effective use of our framework.