NOFE - Neural Operator Function Embedding

Dimensionality reduction methods such as UMAP and t-SNE are central tools for visualising high-dimensional data, but their local-neighborhood objectives can preserve sampling noise while distorting global topology. We show that standard local metrics reward this noise memorisation: top-performing embeddings invent cycles and disconnected islands absent from the data. We introduce a topology-faithfulness benchmark based on noisy manifolds with known homology, tune DiRe against it, and find Pareto-optimal configurations that match or beat GPU-accelerated UMAP on classification while recovering exact first Betti numbers on stress tests. On 723K arXiv paper embeddings, DiRe preserves 3-4 times more topological structure than UMAP at comparable wall-clock.

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.

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.