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arxiv: 2604.21617 · v1 · submitted 2026-04-23 · 💻 cs.CV

Local Neighborhood Instability in Parametric Projections: Quantitative and Visual Analysis

Frederik L. Dennig , Daniel A. Keim This is my paper

Pith reviewed 2026-05-09 22:21 UTC · model grok-4.3

classification 💻 cs.CV
keywords parametric projectionsstability analysisneighborhood instabilityGaussian perturbationsUMAPt-SNEprojection qualityvisual analytics
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The pith

A stability framework using Gaussian perturbations identifies unstable regions in parametric projections missed by reconstruction error and neighborhood metrics.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops a framework to test local stability in parametric projections such as neural-network versions of UMAP and t-SNE. It selects anchor points, adds Gaussian noise to them, and tracks how the 2D neighborhoods around those points shift in the embedding. Quantitative scores for displacement, bias, and assignment error are paired with visualizations of vectors, local PCA shapes, and Voronoi cells. When applied to MNIST and Fashion-MNIST, the method flags unstable areas that standard reconstruction and neighborhood-preservation checks do not reveal, which matters for analysts who embed new points in real time under noisy or drifting data.

Core claim

By probing parametric projections with Gaussian perturbations around selected anchor points and assessing how neighborhoods deform in the 2D embedding, the framework 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, demonstrating that it detects unstable projection regions invisible to reconstruction error or neighborhood-preservation metrics.

What carries the argument

Stability evaluation framework that applies Gaussian perturbations to anchor points and measures resulting 2D neighborhood deformation through both numerical scores and targeted visualizations.

If this is right

  • Jacobian regularization reduces instability in the tested neural projectors.
  • The framework works on both UMAP- and t-SNE-based projectors across different network sizes.
  • Visual tools such as displacement vectors and Voronoi misassignment maps allow detailed inspection of specific unstable locations.
  • Standard reconstruction error and neighborhood-preservation metrics can miss local instabilities that the perturbation approach reveals.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same perturbation-and-visualization approach could be tested on non-parametric projection methods to compare their local robustness.
  • Unstable regions might correlate with particular data characteristics such as class boundaries or low-density areas, suggesting targeted improvements to the projectors.
  • In deployed systems the framework could generate real-time warnings when new points fall near flagged unstable zones.

Load-bearing premise

That Gaussian perturbations around selected anchor points adequately represent the real input variations from measurement noise or data drift that occur in practice.

What would settle it

If regions the framework labels unstable show no greater displacement under actual recorded measurement noise or data drift than regions it labels stable, the claim that the framework detects meaningful instabilities would be falsified.

Figures

Figures reproduced from arXiv: 2604.21617 by Daniel A. Keim, Frederik L. Dennig.

Figure 1
Figure 1. Figure 1: Framework overview on MNIST. (1) A baseline UMAP projection is computed. (2) Class centroid-based anchors are selected. (3) Isotropic Gaussian noise is added to perturb anchor samples. (4) A multi-layer perceptron (MLP) projects all noisy samples into 2D. (5) Stability is assessed with three metrics: Mean displacement (Ddev), displacement bias (Dbias), and nearest-anchor assignment error (ENA); lower is be… view at source ↗
Figure 2
Figure 2. Figure 2: Top – Class centroid-based anchors of MNIST. Bottom – Same images with isotropic Gaussian noise added (σ = 0.17). centroids in projection space ensures representative anchors rather than outliers. (3) We generate perturbed inputs around each anchor by adding isotropic Gaussian noise. (4) We project the perturbed samples and compute quantitative stability metrics. (5) We visualize the resulting point clouds… view at source ↗
Figure 3
Figure 3. Figure 3: Qualitative stability comparison across UMAP and t-SNE projections. Each row shows the reference projection with anchors, the most and least stable MLP by mean displacement (Tab.2) with stability visualizations. MNIST/Fashion is ∼135 ms for MLP-large, dominated by MLP forward passes; ENA is O(NA2 ) and sub-millisecond. Qualitative Results [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript claims to present a stability evaluation framework for parametric projections (UMAP and t-SNE neural projectors) that probes local neighborhoods with Gaussian perturbations around anchor points. It computes quantitative measures including mean displacement, bias, and nearest-anchor assignment error, paired with visualizations of displacement vectors, local PCA ellipsoids, and Voronoi misassignments. Experiments on MNIST and Fashion-MNIST show identification of unstable regions not visible to reconstruction error or neighborhood-preservation metrics, and evaluate Jacobian regularization.

Significance. If the results hold, this provides a valuable tool for analyzing and improving the robustness of parametric embeddings, which are key for interactive data visualization. The combination of quantitative metrics and visual inspection on standard datasets is a strength, offering falsifiable and inspectable findings. It could help address issues with data drift in deployed projection systems.

major comments (2)
  1. [§3 (stability evaluation framework, perturbation model)] §3 (stability evaluation framework, perturbation model): The central claim that the framework identifies unstable regions invisible to reconstruction error or neighborhood-preservation metrics rests on isotropic Gaussian perturbations adequately representing real input variations. No cross-validation or ablation against structured perturbations (e.g., small rotations, translations, or intensity shifts typical for MNIST/Fashion-MNIST) is reported, so the practical relevance of the detected instability does not follow from the experiments.
  2. [§5 (results, quantitative measures)] §5 (results, quantitative measures): The reported values for displacement, bias, and assignment error lack error bars, standard deviations across random seeds, or statistical significance tests. This makes it difficult to assess whether the differences between regions or the effect of Jacobian regularization are reliable, weakening support for the claim of newly identified unstable areas.
minor comments (2)
  1. [Abstract and §4] The abstract and methods could more explicitly list the network architecture sizes and training hyperparameters used for the neural projectors.
  2. [Figures] Figure captions for the per-anchor visualizations should include details on how the local PCA ellipsoids and Voronoi cells are scaled.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive feedback on our manuscript. We address each major comment below and describe the revisions we will incorporate to improve the work.

read point-by-point responses

  1. Referee: §3 (stability evaluation framework, perturbation model): The central claim that the framework identifies unstable regions invisible to reconstruction error or neighborhood-preservation metrics rests on isotropic Gaussian perturbations adequately representing real input variations. No cross-validation or ablation against structured perturbations (e.g., small rotations, translations, or intensity shifts typical for MNIST/Fashion-MNIST) is reported, so the practical relevance of the detected instability does not follow from the experiments.

    Authors: We agree that the perturbation model is central to interpreting the practical relevance of the detected instabilities. Isotropic Gaussian noise was chosen as a controlled, standard model for small additive variations, allowing isolation of local sensitivity without confounding factors. The framework itself is perturbation-agnostic and the quantitative measures (displacement, bias, assignment error) are defined independently of the specific noise distribution. We acknowledge that ablations with structured perturbations would provide stronger evidence for real-world applicability. In the revised manuscript we will expand the discussion in §3 to explicitly address this limitation, note that the current results demonstrate the framework's utility under Gaussian perturbations, and add a small ablation using intensity shifts on MNIST to illustrate consistency with at least one structured variation. revision: partial

  2. Referee: §5 (results, quantitative measures): The reported values for displacement, bias, and assignment error lack error bars, standard deviations across random seeds, or statistical significance tests. This makes it difficult to assess whether the differences between regions or the effect of Jacobian regularization are reliable, weakening support for the claim of newly identified unstable areas.

    Authors: We accept that the absence of variability measures limits the strength of the quantitative claims. The original runs used fixed random seeds for reproducibility across network sizes and regularization settings. In the revision we will recompute the key metrics (mean displacement, bias, nearest-anchor error) over multiple independent seeds, report standard deviations, and add error bars to the relevant figures and tables in §5. We will also include a short statistical comparison (e.g., paired t-tests) between regions and between regularized vs. unregularized models to support the reliability of the observed differences. revision: yes

Circularity Check

0 steps flagged

No significant circularity in the stability evaluation framework

full rationale

The paper defines its core quantitative measures (mean displacement, bias, and nearest-anchor assignment error) as direct computations performed on the 2D positions obtained after applying Gaussian perturbations to selected anchor points. These quantities are calculated independently from the perturbed embeddings and are not defined in terms of each other, nor are they obtained by fitting parameters that are then renamed as predictions. The comparison against reconstruction error and neighborhood-preservation metrics relies on standard external definitions rather than any self-referential construction or self-citation chain. No equations or steps in the described framework reduce to tautological inputs by construction, making the derivation self-contained.

Axiom & Free-Parameter Ledger

2 free parameters · 1 axioms · 0 invented entities

The framework rests on the modeling choice that isotropic Gaussian noise captures relevant input variations and on the assumption that the trained parametric projectors behave as black-box functions whose local behavior can be probed by finite perturbations.

free parameters (2)
  • Gaussian perturbation variance
    Chosen value controls the scale of input noise; not derived from data.
  • Network architecture sizes
    Different hidden-layer widths are tested; selection affects measured stability.
axioms (1)
  • domain assumption Gaussian perturbations adequately model measurement noise and data drift
    Invoked when the framework is presented as a probe for real-world input variations.

pith-pipeline@v0.9.0 · 5456 in / 1179 out tokens · 31328 ms · 2026-05-09T22:21:52.731565+00:00 · methodology

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