Local Neighborhood Instability in Parametric Projections: Quantitative and Visual Analysis

2026-04-23

AI Summary

This work addresses the instability of parametric projections under input perturbations and the lack of effective evaluation metrics for local neighborhood stability. We propose a systematic assessment framework based on Gaussian perturbation probes, which— for the first time—integrates quantitative measures including displacement mean, variance, and nearest-anchor assignment error. Complementing these metrics, we introduce fine-grained visual diagnostics through displacement vectors, local PCA ellipsoids, and Voronoi misassignment maps. Evaluated on MNIST and Fashion-MNIST, our approach successfully identifies unstable regions that conventional reconstruction errors and neighborhood preservation metrics fail to detect, thereby establishing a new paradigm for robustness evaluation of projection methods.

Technology Category

• Computer Vision: Adversarial Attacks & Robustness
• Machine Learning: Calibration & Uncertainty Quantification
• Natural Language Processing: Safety and Robustness

Application Category

• Security and Privacy: Large-scale security measurements
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics

Abstract

Parametric projections let analysts embed new points in real time, but input variations from measurement noise or data drift can produce unpredictable shifts in the 2D layout. Whether and where a projection is locally stable remains largely unexamined. In this paper, we present a stability evaluation framework that probes parametric projections with Gaussian perturbations around selected anchor points and assesses how neighborhoods deform in the 2D embedding. Our approach combines quantitative measures of mean displacement, bias, and nearest-anchor assignment error with per-anchor visualizations of displacement vectors, local PCA ellipsoids, and Voronoi misassignment for detailed inspection. We demonstrate the framework's effectiveness on UMAP- and t-SNE-based neural projectors of varying network sizes and study the effect of Jacobian regularization as a gradient-based robustness strategy. We apply our framework to the MNIST and Fashion-MNIST datasets. The results show that our framework identifies unstable projection regions invisible to reconstruction error or neighborhood-preservation metrics.

Problem

• local neighborhood instability
• parametric projections
• data drift
• measurement noise
• embedding stability

Innovation

• local stability
• parametric projections
• Gaussian perturbations
• neighborhood deformation
• Jacobian regularization