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 X to the embedding 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 Pij from paired input-space views. Binary cross-entropy matches these targets to cross-view Student t -kernel affinities Qij over selected hard pairs. (C) The density-preservation loss compares kNN log-density profiles gsX and gsY across anchor observations at scales s∈{⌊k/2⌋,k} . Stop-gradient is applied to the input-space profile. (D) The final objective is L(θ)=Lmanifold(θ)+λdLdens(θ) ; its gradients update the encoder and projection module.
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 ± 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 , ≈698 K points) benchmark.
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.
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
High-dimensional biomedical data, such as cell-by-gene matrices, are increasingly generated temporally. However, Manifold Learning algorithms, like t-SNE and UMAP, cannot incorporate time-ordering in their layouts, obfuscating the dynamics of cell types or other classes. As a solution, we present IRIS, a new Manifold Learning algorithm that structures layouts both chronologically and by manifold topology. IRIS can visualize a wide range of dynamic biomedical data, including scRNA-seq, comparative metagenomics, and literature.
Brian Ondov, Chia-Hsuan Chang, Weipeng Zhou +6
Department of Biomedical Informatics and Data Science, Yale School of Medicine, 333 Cedar St, New Haven, 06510, CT, USA. · School of Information, University of Michigan, 500 S. State St, Ann Arbor, 48109, MI, USA.
Low-dimensional embeddings are widely used as visual summaries of high-dimensional data and to enable downstream scientific discoveries. Yet, popular nonlinear dimension reduction methods, such as t-SNE and UMAP, are often selected based on visual appeal alone and without rigorous quantitative validation. A major reason is that manifold embeddings typically do not provide an out-of-sample map nor an inverse back to the original feature space; this makes held-out validation, the gold standard in supervised learning, all but impossible. To address these challenges, we develop a novel framework, MEDAL (Manifold Embedding Distillation via Autoencoder Learning), which distills a fitted manifold embedding into a reusable encoder--decoder model. MEDAL trains a constrained autoencoder whose bottleneck exactly matches any teacher embedding while the decoder reconstructs the original input; this yields an explicit map for new samples, an approximate inverse, and a pointwise reconstruction-based measure of distortion in the manifold space. This converts static manifold embeddings into models that can be evaluated on held-out data, enabling quantitative validation including comparing different dimension reduction methods as well as hyperparameter tuning. Across multiple benchmark and scientific case studies, we show that MEDAL enables held-out validation to determine optimal manifold embeddings and hyperparameters, reveals biologically coherent regions that are difficult to preserve in two dimensional embeddings, and detects distribution shift when new samples are mapped into a fixed reference manifold. MEDAL provides a general validation wrapper to any existing dimension reduction technique that will improve the rigor and reliability of dimension reduction in scientific workflows.
Irene Chang, Tarek M. Zikry, Genevera I. Allen
Department of Statistics, Columbia University, New York, NY 10027 · School of Data and Information Sciences, University of North Carolina at Chapel Hill, Chapel Hill, NC 27599 · Irving Institute for Cancer Dynamics, Columbia University, New York, NY 10027 +1