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This paper introduces a framework for evaluating the local stability of parametric projections by probing them with Gaussian perturbations around anchor points and quantifying the resulting deformations in the 2D embedding space. The framework combines quantitative metrics like mean displacement and nearest-neighbor error with visualizations of displacement vectors and PCA ellipsoids to provide a detailed assessment of stability. Experiments on UMAP and t-SNE neural projectors trained on MNIST and Fashion-MNIST demonstrate that the framework can identify unstable regions missed by traditional metrics, highlighting the importance of local stability analysis.
Parametric projections, like UMAP and t-SNE, can have surprisingly unstable local neighborhoods, leading to unpredictable shifts in the 2D layout even with small input variations.
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.
