分析参数化投影的局部不稳定性，揭示传统指标遗漏的异常区域。
Local Neighborhood Instability in Parametric Projections: Quantitative and Visual Analysis
- 通过高斯扰动检测锚点周围邻域变形，量化位置偏移与近邻错误
- 发现投影不稳定区在重建误差指标下仍隐藏，且受网络规模影响
- 提供可视化工具定位问题区域，适合模型调试与可解释性研究
参数化投影支持实时嵌入新数据点，但测量噪声或数据漂移引发的输入变化会导致二维布局不可预测地偏移。投影是否及在何处具有局部稳定性尚未被充分研究。本文提出一种稳定性评估框架，对选定锚点施加高斯扰动，分析其邻域在二维嵌入中的形变情况。方法结合均值位移、偏差和最近锚点分配误差等定量指标，并辅以每锚点的位移向量图、局部PCA椭球与Voronoi误分配可视化，实现细致检查。我们在不同规模的UMAP与t-SNE神经投影器上验证了该框架的有效性，考察了雅可比正则化作为梯度基鲁棒策略的效果。实验基于MNIST与Fashion-MNIST数据集，结果表明：该框架能识别出重建误差或邻域保持度量无法察觉的不稳定区域。
原文摘要 · Abstract (English)
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.
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