DR-SNE: Density-Regularized Stochastic Neighbor Embedding

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.

Dimensionality reduction (DR) involves two longstanding trade-offs. First, preserving local neighborhoods can come at the cost of global structure. Neighbor embedding methods such as t-SNE and UMAP prioritize local similarity preservation but do not explicitly constrain global organization, whereas standard spectral methods such as Laplacian Eigenmaps capture smooth, coarse-scale graph structure but offer limited flexibility to depict finer local structure. Second, the flexibility of nonlinear DR methods often comes at the cost of analytical transparency. Many methods do not explicitly reveal how high-dimensional structure produces patterns in the embedding. We introduce SDMP (Spectral Decomposition for Multiscale Projection), a nonlinear DR framework built on an explicit spectral decomposition. In this formulation, each embedding dimension is expressed as a weighted combination of Laplacian eigenvectors derived from a neighborhood graph, with the weights learned via a UMAP-style cross-entropy objective. By progressively expanding the spectral subspace to capture increasingly fine graph structure, SDMP produces a sequence of embeddings, making the evolving balance between global organization and local detail explicit, controllable, and inspectable. The explicit decomposition also reveals which spectral scales shape the overall embedding and how individual eigenvectors influence point positions. Quantitative evaluations on synthetic, image, and single-cell data show competitive local and global structure preservation, while case studies illustrate how the decomposition supports interpretation of clusters and developmental trajectories across spectral scales.

It is common for two-dimensional embeddings of high-dimensional data to be read far beyond what they can support. Distances in and between clusters, the meaning behind empty spaces, and the amount of structure hidden at each point are generally invisible in the output of methods such as t-SNE and UMAP. This is because the information that could support the meaning of these properties is discarded during the optimisation process. Here, we present FloDR, a dimensionality reduction method that embeds data through an invertible normalising flow. While FloDR only uses the first two output coordinates to create a two-dimensional embedding, it retains the remaining coordinates rather than discarding them. In addition to the embedding, an exact inverse and an exact density are properties of a trained mapping, which enable diagnostic visualisations that are computed from the exact inverse of the model that drew the layout rather than from an approximate one. Specifically, we draw two fields, the conditional spread, which measures how much of the original data remains undetermined at each embedding position in input units, and the hidden contrast, which measures how much information about a labelled contrast the two plotted coordinates discard. Both fields are rendered with a prespecified test against a held out portion of the input data and a bootstrap confidence. A field that fails the test is reported as refused.