DMT-Dens: Density-preserving manifold visualization for biological data

Ruizhe Wang; Yixuan Dong; Bolin Yang; Bingo Wing-Kuen Ling; Fuji Yang; Zelin Zang

q-bio.QMAug 18, 2026

DMT-Dens: Density-preserving manifold visualization for biological data

Authors: Ruizhe Wang, Yixuan Dong, Bolin Yang, Bingo Wing-Kuen Ling, Fuji Yang, Zelin Zang

Organizations: Tsientang Institute for Advanced Study, Hangzhou, Zhejiang, China

Abstract

Motivation: Low-dimensional embeddings are widely used to explore cell-state heterogeneity in single-cell and other high-dimensional biological data. Although many methods preserve local neighborhoods, they may distort the apparent sampling density of processed observations, altering the visual contrast between dense and sparse regions and complicating the interpretation of rare, transitional, or continuous cell-state populations. Results: We present DMT-Dens, a parametric manifold-visualization method built on a latent-token Transformer encoder. The model integrates rank-based manifold alignment with hard-pair aggregation. To preserve density, it optimizes a loss based on the Pearson correlation between k-nearest-neighbor log-radius estimates in the processed input and two-dimensional embedding spaces. Benchmark evaluations demonstrate strong density preservation, particularly on biological datasets, while retaining competitive label separability. Availability: Source code, data-processing scripts, and resolved experiment configurations are available at https://github.com/Ruizhe-wang/DMT-Dens.

Figures & tables

Figure 1: Motivation for density-preserving biological visualization. a, Ground-truth synthetic three-dimensional branching tree with a dense root, sparse transitions, and multiple branch points, colored by branch identity. b, Two-dimensional embeddings of the same data. The displayed baseline embeddings differ in local density, apparent continuity, and adjacency. Adding a density objective to t-SNE and UMAP yields the density-aware baselines den-SNE and densMAP. The red outlines mark apparent disconnections in this representative UMAP embedding and their partial reconnection in densMAP. In the displayed DMT-Dens embedding, density variation and continuous paths are visually retained. c, Density preservation versus label separability across the benchmark datasets. Each point shows the mean rank of one method over the eight datasets with complete baseline coverage (density-correlation rank versus SVC-accuracy rank), and whiskers show the standard error across datasets. The dashed curve marks the Pareto frontier of the baseline methods.

Figure 2: DMT-Dens architecture and training objective. (A) A latent-token Transformer encoder and projection module map the input matrix

\mathbf{X}

to the embedding

\mathbf{Y}

. For each observation, the encoder forms a fixed set of latent tokens, applies self-attention within the token set, and pools the result into a 40-dimensional representation for two-dimensional projection. (B) The manifold-preservation loss derives bidirectional rank affinities

P_{ij}

from paired input-space views. Binary cross-entropy matches these targets to cross-view Student

t

-kernel affinities

Q_{ij}

over selected hard pairs. (C) The density-preservation loss compares kNN log-density profiles

g_{s}^{X}

and

g_{s}^{Y}

across anchor observations at scales

s\in\{\lfloor k/2\rfloor,k\}

. Stop-gradient is applied to the input-space profile. (D) The final objective is

\mathcal{L}(\theta)=\mathcal{L}_{\mathrm{manifold}}(\theta)+\lambda_{d}\mathcal{L}_{\mathrm{dens}}(\theta)

; its gradients update the encoder and projection module.

Panel of Table 1 — Table 1: Main comparison on density preservation and label separability. Higher is better for both metrics, and the best (highest) value in each row is shown in bold . Entries report mean $\pm$ standard deviation over available seeds. The final row of each panel reports the unweighted mean over all nine datasets; this mean is omitted for den-SNE because its EMNIST run did not complete. OOT (out of time) denotes a runtime of 24 h+; this applies to den-SNE on the full EMNIST ( byclass , ${\approx}698$ K points) benchmark.

Dimensionality Reduction

Dimensionality reduction maps high-dimensional data to compact representations for visualization, prediction, compression, and scientific modeling.

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May 28, 2026J/K move · Enter open · S save

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

Rajas Poorna, Marcus T. Cicerone

Georgia Institute of Technology · School of Chemical and Biomolecular Engineering Georgia Institute of Technology · School of Chemistry and Biochemistry Georgia Institute of Technology

Data Visualization Manifold Learning Dimensionality Reduction

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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.

Abstract

Dimensionality Reduction

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