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

Low-dimensional embeddings are essential for visualizing cell-state heterogeneity in single-cell transcriptomics and other biological data. However, standard nonlinear neighbor-embedding methods like t-SNE and UMAP frequently distort the relative sampling density of processed observations, artificially expanding dense regions or compressing sparse ones. This distortion obscures the visual contrast between distinct cell populations and complicates the interpretation of rare cell types, transitional states, or continuous developmental trajectories. This paper asks whether a parametric deep manifold-learning model can be designed to jointly preserve local neighborhood structures and relative observation densities without sacrificing label separability.

The authors introduce DMT-Dens, a parametric manifold-visualization framework built on a latent-token Transformer encoder paired with a projection module. The model maps high-dimensional input observations into a 40-dimensional latent space before projecting them into a two-dimensional embedding. The optimization objective combines two core components:

1. A manifold-preservation loss that integrates bidirectional rank-based affinities, Student t-kernel similarities, and a hard-pair aggregation strategy to ensure that local neighborhood relationships are properly maintained.
2. A density-preservation loss that optimizes the Pearson correlation between k-nearest-neighbor log-radius estimates computed in the processed high-dimensional input space and the two-dimensional embedding space across multiple scales.

The encoder uses a fixed number of latent tokens processed via multi-head self-attention independently of the observation dimension, avoiding the quadratic scaling costs associated with standard transformer architectures.

Evaluated across a benchmark suite comprising four biological datasets (including single-cell atlases and epithelial states) alongside image, text, and synthetic datasets, DMT-Dens demonstrates superior density preservation compared to existing visualization baselines. Specifically, DMT-Dens achieves the highest density correlation across all four biological datasets and six of the nine total datasets. At the same time, it maintains competitive label separability, ranking among the top two methods on seven datasets using linear support vector classification accuracy. Ablation experiments confirm that the explicit density-preservation loss is the primary driver of this improved density fidelity.