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arxiv: 2605.00637 · v1 · submitted 2026-05-01 · 💻 cs.LG

Recognition: unknown

Class Angular Distortion Index for Dimensionality Reduction

Kaviru Gunaratne , Stephen Kobourov , Jacob Miller
Authors on Pith no claims yet

Pith reviewed 2026-05-09 20:25 UTC · model grok-4.3

classification 💻 cs.LG
keywords dimensionality reductioncluster qualityangular distortionCADIprojection evaluationdifferentiable metrict-SNEUMAP
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The pith

The Class Angular Distortion Index measures how faithfully dimensionality reduction projections preserve cluster organization by computing internal angles from point triples.

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

Dimensionality reduction techniques such as t-SNE and UMAP frequently produce projections in which clusters appear distinct yet sit in arbitrary or misleading relative positions. The paper introduces the Class Angular Distortion Index to evaluate the faithfulness of those arrangements through the internal angles formed by triples of points. This approach avoids assumptions about cluster shape or density and supplies usable signals in both real and synthetic cases where standard cluster metrics return uninformative results. Because the index rests on angle calculations, it is differentiable and therefore usable as an objective for training new projection methods, which the authors demonstrate with one such technique.

Core claim

The authors claim that the relative organization of clusters in a projection can be assessed by tracking how internal angles among point triples change from the original space. They formalize this idea as the Class Angular Distortion Index and show that it returns interpretable values on data sets where existing separability or shape-based metrics do not. The same angle foundation makes the index differentiable, allowing it to guide optimization of dimensionality reduction directly toward better cluster arrangements.

What carries the argument

The Class Angular Distortion Index (CADI), which quantifies distortion of internal angles formed by point triples to judge cluster arrangement faithfulness in a projection.

If this is right

  • Projections can be optimized directly against CADI to reduce misleading cluster placements.
  • The index supplies concrete, interpretable feedback on which inter-cluster angles are most distorted.
  • Evaluation of existing methods such as t-SNE and UMAP becomes possible even when clusters are non-spherical or unevenly dense.
  • A new dimensionality reduction algorithm that minimizes CADI is presented as a working example.

Where Pith is reading between the lines

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

  • CADI could be added to routine DR evaluation toolkits alongside neighborhood-preservation scores.
  • Its differentiability opens the possibility of hybrid objectives that balance local structure with global cluster geometry.
  • Different strategies for choosing which point triples to measure may affect sensitivity to particular types of distortion.

Load-bearing premise

Angles computed from point triples in the projected space reliably reflect the true relative organization of clusters in the original data.

What would settle it

A controlled synthetic data set whose ground-truth clusters are rotated or reordered in the projection; CADI should report high distortion while visual inspection or label comparison confirms the change.

Figures

Figures reproduced from arXiv: 2605.00637 by Jacob Miller, Kaviru Gunaratne, Stephen Kobourov.

Figure 1
Figure 1. Figure 1: An example of a synthetic dataset where the globularity assumption for clusters fails to hold. (a) shows the input 3D dataset, which [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: (a) For different numbers of triplets sampled, we calculate CADI for the t-SNE projection of the MNIST dataset on 10,000 different [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Projections of the rings (top) and concentric3 (bottom) datasets created by different DR algorithms. scores to the Random projection, as it assumes that all clusters fully overlap due to the fact that the cluster centroids almost coincide. Other synthetic data. We note that this pattern continues in the remaining two synthetic datasets. UMAP, t-SNE, and PaCMAP continue to separate clusters into uninformati… view at source ↗
Figure 4
Figure 4. Figure 4: Projections of the liver (top), usps (middle), and trec (bottom) datasets created by different DR algorithms. identifies and rewards these clusters formed by t-SNE. However, it fails to recognize the class-level information revealed by AngleEm￾bedding and ranks it last as a result. Other data. Across all datasets, only CADI consistently assigns the worst score to the Random projection. All other metrics at… view at source ↗
Figure 5
Figure 5. Figure 5: CADI scores for UMAP projections of the concentric3 [PITH_FULL_IMAGE:figures/full_fig_p010_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Times taken to compute each metric on the t-SNE projec [PITH_FULL_IMAGE:figures/full_fig_p016_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: For different numbers of triplets sampled, we calculate [PITH_FULL_IMAGE:figures/full_fig_p019_7.png] view at source ↗
Figure 9
Figure 9. Figure 9: The per-pixel standard deviation across images within [PITH_FULL_IMAGE:figures/full_fig_p020_9.png] view at source ↗
Figure 8
Figure 8. Figure 8: The epsilon graphs of the liver dataset are plotted using [PITH_FULL_IMAGE:figures/full_fig_p020_8.png] view at source ↗
Figure 11
Figure 11. Figure 11: AngleEmbedding projection of the TREC dataset. [PITH_FULL_IMAGE:figures/full_fig_p021_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: AngleEmbedding projection of the Emotion dataset. (a) [PITH_FULL_IMAGE:figures/full_fig_p021_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: CADI scores for UMAP projections of the concentric3 dataset with different values for the hyperparameters [PITH_FULL_IMAGE:figures/full_fig_p022_13.png] view at source ↗
read the original abstract

Dimensionality reduction (DR) techniques are often characterized by whether they preserve global, high-level structures in the data or local, neighborhood structures. This distinction matters in visualization: global methods can obscure clusters while local methods can over-emphasize them. Yet, even when clusters appear distinct, their relative arrangement in the projection may be arbitrary or misleading, a common issue in techniques such as t-SNE and UMAP. Existing cluster quality metrics either only measure cluster separability or assume spherical, globular clusters in the original space. We introduce the Class Angular Distortion Index (CADI), a metric that uses internal angles among point triples to determine the faithfulness of cluster organization in a projection. We show cases on both real and synthetic data where existing cluster metrics fail, but CADI provides an interpretable result. Since it relies on computing angles, CADI is also differentiable, enabling optimization. We demonstrate this with a CADI-based DR technique.

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 introduces the Class Angular Distortion Index (CADI), a metric that computes internal angles among point triples in a low-dimensional projection to assess the faithfulness of cluster organization. The authors claim CADI succeeds on real and synthetic data where existing cluster metrics fail (due to assumptions of spherical clusters or focus only on separability), requires no assumptions on cluster shape, density, or triple selection, is differentiable (enabling optimization), and is demonstrated via a CADI-optimized DR technique.

Significance. If the central claims hold, CADI would provide a useful, interpretable addition to DR evaluation tools by addressing limitations of metrics like silhouette score or Davies-Bouldin index for non-globular clusters. The differentiability property is a concrete strength that could support new optimization-based DR methods, and the empirical cases where standard metrics fail illustrate potential practical value.

major comments (2)
  1. [Abstract and §3] Abstract and §3 (definition of CADI): the claim that CADI 'works without needing assumptions about cluster shape, density, or the specific choice of point triples' is load-bearing but unsupported. The metric is defined from angles on point triples, yet no explicit, parameter-free rule for selecting or averaging over the triple set is given; random, nearest-neighbor, or density-biased sampling would systematically affect results for elongated, non-convex, or varying-density clusters—the exact regimes highlighted as failures of prior metrics.
  2. [§4] §4 (differentiability and optimization): the assertion that CADI is differentiable and enables a new DR technique is central to the contribution, but no explicit derivative, gradient expression, or verification that the angle-based formulation remains well-defined under projection is supplied, leaving the optimization demonstration unverifiable.
minor comments (2)
  1. [§3] Notation for the angle computation and the final CADI formula is introduced without a clear preliminary definition or diagram, making the geometric construction harder to follow on first reading.
  2. [Abstract] The abstract states the metric 'exists and is differentiable' but supplies no equations; moving a concise definition or key formula to the abstract would improve accessibility.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive and detailed review. The two major comments identify areas where the manuscript would benefit from greater explicitness. We address each point below and will revise the manuscript to incorporate the requested clarifications and mathematical details.

read point-by-point responses

  1. Referee: [Abstract and §3] Abstract and §3 (definition of CADI): the claim that CADI 'works without needing assumptions about cluster shape, density, or the specific choice of point triples' is load-bearing but unsupported. The metric is defined from angles on point triples, yet no explicit, parameter-free rule for selecting or averaging over the triple set is given; random, nearest-neighbor, or density-biased sampling would systematically affect results for elongated, non-convex, or varying-density clusters—the exact regimes highlighted as failures of prior metrics.

    Authors: We agree that the current text does not supply an explicit rule for triple selection and averaging. We will revise the abstract and §3 to state that CADI is defined by averaging the absolute difference between the original-space and projection-space angles over all possible triples of points that belong to the same class. This choice is parameter-free and does not rely on cluster shape, density, or any sampling heuristic. We will also add a short robustness subsection demonstrating that the resulting index remains stable under moderate subsampling for the non-globular and varying-density cases already shown in the experiments. revision: yes

  2. Referee: [§4] §4 (differentiability and optimization): the assertion that CADI is differentiable and enables a new DR technique is central to the contribution, but no explicit derivative, gradient expression, or verification that the angle-based formulation remains well-defined under projection is supplied, leaving the optimization demonstration unverifiable.

    Authors: We acknowledge that §4 currently asserts differentiability without supplying the gradient. In the revision we will insert the explicit gradient of CADI with respect to the low-dimensional coordinates, obtained by differentiating the arccos expression for each triple angle via the chain rule on vector dot products and norms. We will also note the measure-zero set of configurations where points are collinear (where the derivative is undefined) and describe how the optimization procedure handles these cases. A brief verification that the gradient is well-defined for generic projections will be included. revision: yes

Circularity Check

0 steps flagged

CADI is introduced as a direct geometric definition with no reduction to fitted inputs or self-citations.

full rationale

The paper defines the Class Angular Distortion Index explicitly from internal angles computed on point triples in the projection. This is a first-principles construction using standard Euclidean geometry rather than any fitted parameter, renamed empirical pattern, or load-bearing self-citation. No equations or claims in the abstract reduce the metric to its own inputs by construction; the differentiability claim follows immediately from the angle formula. The paper contrasts CADI with existing metrics on real and synthetic data without invoking prior author work as a uniqueness theorem or ansatz. The derivation chain is therefore self-contained as a new index definition.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central claim rests on the geometric interpretation of angles as faithful indicators of cluster organization and on the assumption that this measure is meaningfully different from existing metrics.

axioms (1)
  • domain assumption Euclidean angles computed from point coordinates capture the intended notion of cluster angular organization.
    Invoked when the metric is defined from internal angles among point triples.
invented entities (1)
  • Class Angular Distortion Index (CADI) no independent evidence
    purpose: To quantify faithfulness of cluster angular arrangement in projections.
    Newly introduced quantity whose independent validation is not shown in the abstract.

pith-pipeline@v0.9.0 · 5460 in / 1162 out tokens · 24911 ms · 2026-05-09T20:25:32.150714+00:00 · methodology

discussion (0)

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Reference graph

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