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

2025-07-22

AI Summary

UMAP projections suffer from poor stability and weak structural interpretability due to stochastic optimization—particularly sensitivity to initial point placement and negative sampling. To address this, we propose the first (r,d)-stability quantification framework specifically designed for UMAP. Our method systematically evaluates how initial projection perturbations and negative sampling affect neighborhood preservation by constructing “ghost” replicas of data points. We further introduce an adaptive pruning strategy that enhances computational efficiency while preserving detection sensitivity, and develop an interactive visualization tool for diagnostic analysis. Extensive evaluation on multiple real-world datasets demonstrates that our approach efficiently identifies approximately 90% of randomness-dominated unstable points, reducing runtime by 60% compared to baseline methods. This work provides both theoretical foundations and practical tools to enhance the reliability, interpretability, and trustworthiness of UMAP-based dimensionality reduction.

Technology Category

• Machine Learning: Dimensionality Reduction/Feature Selection
• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Sampling/Simulation-based Search

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking

Abstract

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.

Problem

• Analyzing UMAP's stochastic optimization impact on results
• Introducing (r,d)-stability to measure projection reliability
• Developing tools to assess and visualize UMAP stability

Innovation

• Introduces (r,d)-stability framework for UMAP
• Uses ghost duplicates to analyze stochastic effects
• Develops adaptive dropping for efficient computation