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

• URL: http://arxiv.org/abs/2605.30597v1
• Date: Thu, 28 May 2026 21:44:17 GMT
• Title: ScaleMAP: Preserving Local Density and Neighborhood Structure in Low-Dimensional Embeddings
• Abstract summary: DensMAP and PaCMAP adaptively normalize local distances during graph construction, erasing neighborhood scale from the data.<n> ScaleMAP takes a different approach: each pairwise embedding displacement is divided by the geometric mean of the two endpoints' original-space local radii.<n>Across standard benchmarks and scientific datasets from transcriptomics, hyperspectral imaging, and flow, ScaleMAP matches DensMAP on density preservation while maintaining UMAP-level neighborhood.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• 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.

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