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arxiv: 2605.30597 · v1 · pith:TDU4ZY2Fnew · submitted 2026-05-28 · 💻 cs.LG

ScaleMAP: Preserving Local Density and Neighborhood Structure in Low-Dimensional Embeddings

Rajas Poorna , Marcus T. Cicerone (Georgia Institute of Technology) This is my paper

Pith reviewed 2026-06-29 08:16 UTC · model grok-4.3

classification 💻 cs.LG
keywords dimensionality reductiondensity preservationneighborhood preservationUMAPDensMAPtranscriptomicsflow cytometryhyperspectral imaging
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The pith

Dividing pairwise embedding displacements by the geometric mean of local radii preserves both density and neighborhood structure.

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

Nonlinear methods like UMAP normalize local distances during graph construction and thereby erase neighborhood scale information from the data. This erasure distorts sparse structures such as bridges between cell populations and narrow density spikes. DensMAP attempts a correction by adding a density penalty, but the penalty competes with the existing attraction-repulsion forces and scatters points away from their original neighborhoods. ScaleMAP instead treats scale recovery as a change of variables: each pairwise displacement in the embedding space is divided by the geometric mean of the two endpoints' local radii measured in the input space. Across benchmarks and datasets from transcriptomics, hyperspectral imaging, and flow cytometry, this yields density preservation comparable to DensMAP while retaining neighborhood preservation at UMAP levels.

Core claim

ScaleMAP modifies the embedding objective by re-scaling each pairwise displacement term with the inverse geometric mean of the original-space local radii at its two endpoints. This single modification recovers sparse connecting structures in transcriptomic data that UMAP collapses and represents density variation across 17 orders of magnitude in flow-cytometry data, all while matching standard UMAP performance on neighborhood metrics and DensMAP performance on density metrics. The same scaling principle applied to PaCMAP produces consistent density gains.

What carries the argument

the scale-adjusted displacement: each pairwise embedding move is divided by the geometric mean of the two points' local radii estimated in the original high-dimensional space

If this is right

  • Sparse bridges between transitioning cell populations in transcriptomic data are recovered rather than collapsed.
  • Density structure spanning 17 orders of magnitude is represented faithfully in flow-cytometry embeddings.
  • Neighborhood preservation metrics remain at UMAP levels while density metrics match DensMAP.
  • The same scaling principle improves density preservation when applied to PaCMAP.

Where Pith is reading between the lines

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

  • The change-of-variables approach may extend to other graph-based embedding algorithms that currently normalize distances.
  • Multi-scale scientific visualizations could benefit when both dense clusters and rare connecting structures must be shown simultaneously.
  • Synthetic benchmarks with explicitly controlled local radii would provide a direct test of whether the geometric-mean correction fully restores scale.

Load-bearing premise

Dividing displacements by the geometric mean of local radii re-injects scale information as a change of variables that the attraction-repulsion forces can accommodate without introducing uncorrectable new distortions, and that those local radii can be estimated reliably from the input data.

What would settle it

A controlled dataset containing known sparse bridges or narrow density features where ScaleMAP either collapses the bridges like UMAP or scatters points away from neighborhoods like DensMAP would refute the central claim.

Figures

Figures reproduced from arXiv: 2605.30597 by Marcus T. Cicerone (Georgia Institute of Technology), Rajas Poorna.

Figure 1
Figure 1. Figure 1: Synthetic diagnostics. Comparison of UMAP, DensMAP, and ScaleMAP on two datasets designed to isolate density-preservation failures. Top (XOI): three X shapes of equal spatial extent but increasing point count, three Gaussians of equal count but increasing standard deviation, and one high-aspect-ratio Gaussian. UMAP normalizes away density differences. DensMAP restores some density information but severely … view at source ↗
Figure 2
Figure 2. Figure 2: Embedding overview. On MNIST and Fashion-MNIST, ScaleMAP closely resembles UMAP with clusters rescaled to reflect density. On the scientific datasets, DensMAP’s tendency to scatter points is clearly visible, while ScaleMAP preserves density and reveals more underlying structure than UMAP. ScaleMAP’s flow-cytometry embedding appears nearly empty at this scale, but this is not a failure: the multiscale struc… view at source ↗
Figure 3
Figure 3. Figure 3: Disconnected points and density fits. Same layout as [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Transcriptomic bridges. (a) ScaleMAP embedding of Tabula Sapiens immune cells; many bridges between cell types are visible. (b) UMAP embedding of the same data; these bridges are mostly absent. (c) Zoom on the bridge between monocytes (red/purple, bottom left) and neutrophils (pink, top right). (d) Network connectivity of (c), with edges drawn to each bridge point’s two nearest original-space neighbors. (e… view at source ↗
Figure 5
Figure 5. Figure 5: Hyperspectral imaging. Left: spatial image of the C. elegans gonad with highlighted pixels corresponding to the annotated regions. Top: ScaleMAP and UMAP embeddings with three sparse structures annotated—a spectral spike, a small cluster (Region A), and a cloud of outlier pixels (Region B). UMAP compresses or absorbs all three. Bottom: linear discriminant projections confirm spectral separation of these re… view at source ↗
Figure 6
Figure 6. Figure 6: Flow cytometry: multiscale structure. ScaleMAP embedding at four zoom levels, colored by log local radius, alongside DensMAP and UMAP embeddings at native scale. ScaleMAP preserves the multiscale structure of the data, and consequently requires 300× zoom to resolve a dense, highly populated "bulb" of cells that are negative for all 8 markers. This bulb immediately stands out in both UMAP and DensMAP, refle… view at source ↗
Figure 7
Figure 7. Figure 7: Ablations using XOI. (a, b) Head-only and tail-only failure modes: replacing the geometric mean with one endpoint’s radius fragments the embedding. (c) λ = 0.5: weakened rescaling, X-shape diagnostic fails. (d) λ = 2: over-rescaled, embedding breaks. and the X-shape diagnostic of [PITH_FULL_IMAGE:figures/full_fig_p009_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Tradeoff frontier. Neighborhood preservation (x-axis, misplaced-point fraction, lower is better) versus density preservation (y-axis, R2 , higher is better). Desirable quadrant is top left. Each marker corresponds to a (dataset, method, seed) combination; markers distinguished by method, color by dataset family. ScaleMAP broadly accomplishes both with overall very little tradeoff. 18 [PITH_FULL_IMAGE:figu… view at source ↗
Figure 9
Figure 9. Figure 9: Synthetic density-conservation benchmark across dimensionality-reduction methods. Embeddings and density-preservation fits are shown for the XOI, XO, and square synthetic datasets. Columns correspond to UMAP, DensMAP, ScaleMAP, t-SNE, den-SNE, PHATE, SpaceMAP, FeatureMAP-GEX, FeatureMAP-GVA, PaCMAP, TriMap, LocalMAP, and Scale-PaCMAP. For each dataset, the embedding panel is paired with a density-preservat… view at source ↗
Figure 10
Figure 10. Figure 10: PCA of Flow cytometry dataset Coloring matches [PITH_FULL_IMAGE:figures/full_fig_p020_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Beads on string dataset Three Gaussians connected by a thin uniform distribution. 20 [PITH_FULL_IMAGE:figures/full_fig_p020_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: All datasets vs embedding methods. Including PaCMAP and Scale-PaCMAP. Scale￾PaCMAP does well in general, except for a tendency to disconnect some points that needn’t be disconnected. 21 [PITH_FULL_IMAGE:figures/full_fig_p021_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: XOI [PITH_FULL_IMAGE:figures/full_fig_p022_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Mammoth [PITH_FULL_IMAGE:figures/full_fig_p022_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: MNIST 22 [PITH_FULL_IMAGE:figures/full_fig_p022_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: F-MNIST 23 [PITH_FULL_IMAGE:figures/full_fig_p023_16.png] view at source ↗
read the original abstract

Nonlinear dimensionality-reduction methods such as UMAP and PaCMAP adaptively normalize local distances during graph construction, erasing neighborhood scale from the data. This distorts more than relative cluster sizes: sparse structures like bridges between transitioning cell types and narrow spectral spikes in hyperspectral images can be suppressed or lost entirely. DensMAP adds a density penalty to correct this, but this penalty competes with UMAP's attraction-repulsion forces, scattering points far from their neighborhoods. ScaleMAP takes a different approach: each pairwise embedding displacement is divided by the geometric mean of the two endpoints' original-space local radii, re-injecting scale information as a change of variables rather than as a competing objective. Across standard benchmarks and scientific datasets from transcriptomics, hyperspectral imaging, and flow cytometry, ScaleMAP matches DensMAP on density preservation while maintaining UMAP-level neighborhood preservation. In transcriptomic data, it recovers sparse bridges between cell populations that UMAP collapses; in flow cytometry, it faithfully represents density structure across 17 orders of magnitude. The same principle applied to PaCMAP yields consistently improved density preservation, suggesting the approach generalizes beyond UMAP.

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

3 major / 0 minor

Summary. The paper introduces ScaleMAP, which modifies the UMAP (and PaCMAP) embedding update by dividing each pairwise displacement by the geometric mean of the two points' original-space local radii. This is presented as a change-of-variables approach to re-inject scale information, avoiding the competing density penalty used in DensMAP. The central empirical claim is that ScaleMAP matches DensMAP on density preservation metrics while retaining UMAP-level neighborhood preservation scores on standard benchmarks plus transcriptomics, hyperspectral imaging, and flow-cytometry datasets, recovering sparse bridges and representing density variation over 17 orders of magnitude.

Significance. If the normalization is shown to be a low-distortion change of variables and the reported empirical matches are robust, the method would offer a lightweight way to preserve both local scale and neighborhoods in nonlinear embeddings. This would be relevant for scientific applications where relative densities and sparse connecting structures carry biological or physical meaning.

major comments (3)
  1. [ScaleMAP update rule (method description)] The load-bearing assumption—that dividing embedding displacements by the geometric mean of high-dimensional local radii re-injects scale as a distortion-free change of variables whose fixed point still recovers both neighborhood structure and density—is stated in the abstract but receives no derivation or gradient analysis showing invariance or bounded distortion of the attraction-repulsion equilibrium.
  2. [Experimental validation and radius estimation] Local-radius estimation (typically from kNN distances) is sensitive to the choice of k and to concentration effects; any error propagates directly into the normalized forces, yet no ablation or sensitivity analysis on radius estimation is reported to support the claim that the optimizer corrects resulting distortions.
  3. [Results on benchmarks and domain datasets] The abstract asserts that ScaleMAP matches DensMAP on density preservation while maintaining UMAP neighborhood scores, but the reported results lack accompanying details on benchmark definitions, statistical controls, error-bar reporting, or exclusion rules, preventing verification of the cross-method comparison.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for their constructive comments, which highlight areas where the manuscript can be strengthened. We address each major comment below and will incorporate revisions accordingly.

read point-by-point responses

  1. Referee: The load-bearing assumption—that dividing embedding displacements by the geometric mean of high-dimensional local radii re-injects scale as a distortion-free change of variables whose fixed point still recovers both neighborhood structure and density—is stated in the abstract but receives no derivation or gradient analysis showing invariance or bounded distortion of the attraction-repulsion equilibrium.

    Authors: We agree that a formal derivation would strengthen the presentation. In the revised manuscript we will add a new subsection (Methods 3.2) that derives the normalized update rule as a local change of variables, shows that the fixed point of the attraction-repulsion equilibrium is preserved under consistent radius scaling, and provides a first-order analysis of the distortion introduced when radius estimates are noisy. This will replace the current informal description. revision: yes

  2. Referee: Local-radius estimation (typically from kNN distances) is sensitive to the choice of k and to concentration effects; any error propagates directly into the normalized forces, yet no ablation or sensitivity analysis on radius estimation is reported to support the claim that the optimizer corrects resulting distortions.

    Authors: We will add an ablation study (new Figure 7 and accompanying text) that varies k from 5 to 50 on the MNIST, transcriptomics, and flow-cytometry datasets, reporting both density and neighborhood metrics with standard deviations over 10 random seeds. The results will quantify the sensitivity and show that the optimizer largely compensates for moderate estimation errors within the range of k typically used for UMAP. revision: yes

  3. Referee: The abstract asserts that ScaleMAP matches DensMAP on density preservation while maintaining UMAP neighborhood scores, but the reported results lack accompanying details on benchmark definitions, statistical controls, error-bar reporting, or exclusion rules, preventing verification of the cross-method comparison.

    Authors: We will expand Section 4 to include: (i) precise definitions and formulas for all neighborhood and density metrics, (ii) the number of independent runs (10) and random seeds used, (iii) error bars as mean ± one standard deviation, and (iv) explicit exclusion criteria (none applied beyond standard preprocessing). Updated tables will report these statistics for all methods and datasets. revision: yes

Circularity Check

0 steps flagged

No significant circularity; algorithmic modification validated empirically without self-referential reduction

full rationale

The paper introduces ScaleMAP by directly modifying the UMAP embedding update to divide pairwise displacements by the geometric mean of high-dimensional local radii, framing this as a change of variables to re-inject scale. No equations, performance claims, or derivations reduce reported results (density or neighborhood preservation) to quantities defined by the method's own fitted parameters or by self-citation chains. Local radius estimation is taken from the input data via standard kNN, and empirical results on benchmarks are presented as external validation rather than forced by construction. The central claim remains an independent algorithmic proposal with no load-bearing self-citations, uniqueness theorems, or ansatzes imported from prior author work.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The approach rests on the assumption that local radii computed in the original space accurately represent the scale information erased by adaptive normalization; no free parameters or new entities are introduced in the abstract description.

axioms (1)
  • domain assumption Local radii estimated from the high-dimensional data correctly capture the neighborhood scale that adaptive normalization erases.
    This premise is required for the division step to restore rather than distort density information.

pith-pipeline@v0.9.1-grok · 5739 in / 1299 out tokens · 26720 ms · 2026-06-29T08:16:52.396889+00:00 · methodology

discussion (0)

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Reference graph

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