NFMap: Topographic 2D Manifold Learning with Invertible Normalizing Flows

Héléna Alexandra Gaspar

cs.LGOct 12, 2026

NFMap: Topographic 2D Manifold Learning with Invertible Normalizing Flows

Authors: Héléna Alexandra Gaspar

Organizations: LovingScience 124 City Road, London EC1V 2NX

Abstract

We present NFMap, a dimensionality reduction method for 2D visualisation that extends Generative Topographic Mapping (GTM) with a bijective normalizing flow. Like GTM, NFMap places a fixed mixture grid prior over a 2D latent map, giving a parametric embedding, an explicit likelihood, and the ability to project new points onto an existing map; the likelihood is an effective out-of-distribution signal on chemical datasets. Unlike GTM, whose latent-to-data mapping is an RBF network of bounded flexibility, NFMap's mapping is a normalizing flow, and is therefore fully invertible: any location on the map can be carried back to data space. To alleviate map crowding on large datasets, NFMap additionally supports heavy-tailed Student-t mixture components. NFMap maps data to a latent space of the same dimension, split into a 2-dimensional navigable map and an off-manifold residual. Because the flow is bijective and its objective penalises volume contraction, variance squeezed out of the residual must reappear in the navigable map. The residual scale is the main control between local neighbourhood fidelity and global distance fidelity; its effect depends on the dimensionality of the data and is coupled to the extent of the latent grid.

Figures & tables

	Parametric	Invertible	Exact density	Landscape grid	Student-t
t-SNE	✓ ∗	✗	✗	✗	✓
UMAP	✓ ∗	✗	✗	✗	✗
GTM	✓	✗	✓	✓	✗
FloDR	✓	✓	✓	✗ †	✗
NFMap	✓	✓	✓	✓	✓ ‡

Table 1: Capabilities of each method. Parametric : unseen points can be projected onto an existing map. Landscape grid : the prior places

K

components at fixed, known positions, so any quantity can be aggregated per node and rendered as a continuous background field at a resolution set by

K

. Student-t : the low-dimensional model uses Student-t tails, a standard remedy for crowding (built into t-SNE’s similarity kernel, and available in NFMap as a Student-t mixture prior). UMAP’s kernel, also used by FloDR, is heavy-tailed but not Student-t.

Figure 1: The direction of the $\sigma_{\perp}$ trade-off is set by the data’s geometry. Mean $\pm$ s.d. over datasets and three seeds, grid_bounds $=12$ .

Figure 2: Ablation of the graph attraction ( $\lambda_{g}$ ) and repulsion ( $\lambda_{r}$ ) weights. The graph terms act as opposing forces on the map’s spread relative to the canvas.

Figure 3: Joint sweep on the chemical datasets. Hatched cells violate the containment constraint.

Chemistry
	NFMap		GTM		t-SNE		UMAP
	G	L	G	L	G	L	G	L
BACE	0.37	0.97	0.34	0.96	0.61	0.98	0.30	0.98
BBBP	0.55	0.89	0.71	0.87	0.59	0.91	0.39	0.92
ESOL	0.76	0.85	0.70	0.87	0.57	0.89	0.18	0.91
DRD2	0.36	0.96	0.42	0.95	0.59	0.96	0.23	0.98

Table 2: Each method at its own best setting (means over 3 seeds, CPU). G: global structure (DistCor; geodesic DistCor on the 3-D manifolds). L: local structure (T/C F1). Each sweep is reduced to its Pareto front over (G, L); the reported point is the front’s knee, or its maximum-G point on the manifolds. Best per dataset in bold. NFMap ’s reported maps keep at least 99% of points inside grid_bounds , except the S-curve (94%). Settings in Table B.1 .

Figure 4: DRD2 at both ends of the $\sigma_{\perp}$ dial, against GTM, t-SNE and UMAP.

Figure 5: S-curve. NFMap and baselines at their Table 2 settings.

Figure 6: The prior over the NFMap coordinates, on digits ( $n=600$ , 64 features, coloured by digit). Left: a single Gaussian. Middle: a mixture of 100 components whose means are learned. Right: the same mixture with its means pinned to a fixed hexagonal grid. Crosses mark the mixture means after training: the learned means leave the lattice and gather in the clusters, while the fixed ones stay where they were placed. All three panels at grid_bounds $=6$ and $\sigma_{\perp}=10$ . DistCor, T/C F1 and silhouette under each panel are means over 3 seeds.

Prior	DistCor	T/C F1	Off grid
Digits
Single Gaussian	$0.648\pm 0.002$	$0.963\pm 0.004$	–
Learned mixture	$0.624\pm 0.005$	$0.973\pm 0.002$	0.0%
Fixed grid ( NFMap )	$0.608\pm 0.010$	$0.973\pm 0.003$	0.9%
Chemistry (5 datasets)
Single Gaussian	$0.657\pm 0.090$	$0.848\pm 0.049$	–

Table 3: Prior ablation across datasets and seeds (mean

\pm

sd; 3 seeds for Digits, 5 seeds for the other datasets). Against a single Gaussian the fixed grid gives equal or better T/C F1 and trades some DistCor on Digits and the 3-D manifolds. Off grid: share of points outside grid_bounds (also an average across datasets and seeds). A single Gaussian has no grid, so off-grid is not reported.

Figure 7: Out-of-distribution detection AUC, averaged over the applicable OOD constructions (feature-permuted inputs and matched-moment noise for every dataset; held-out novel classes in addition where there are $\geq$ 3 classes) and three seeds. Dashed line is chance.

Figure 8: Crowding remedies at full dataset scale (single seed). Left: replacing the Euclidean neighbourhood graph with the hierarchical chemical kernel, on the complete DRD2, hERG and EGFR activity sets. Right: the Student-t degrees-of-freedom sweep on EGFR, with the Gaussian prior shown for reference.

Appendix figures & tables3 assets

Supplementary material from the paper’s appendix.

Appendix

Figure A.1: Severed sphere manifold embedding. NFMap and baselines at their Table 2 settings.

Figure A.2: Swiss roll manifold embedding. NFMap and baselines at their Table 2 settings.

	NFMap	t-SNE	UMAP
Chemistry
BACE	0.3 / 24	200	5 / 0.99
BBBP	3 / 24	200	5 / 0
ESOL	0.01 / 24	300	5 / 0.1
DRD2	0.3 / 24	300	5 / 0.5
EGFR	0.3 / 24	300	5 / 0

Appendix

Table B.1 : Hyperparameters selected by the sweep for each entry of Table 2 (the reported Pareto point; not fixed in advance). NFMap :

\sigma_{\perp}

/ grid_bounds, swept over 14

\sigma_{\perp}

values from

10^{-3}

10^{3}

and grid_bounds 12 and 24 (number of grid nodes fixed at 100). t-SNE: perplexity. UMAP: n_neighbors / min_dist. GTM: not swept, one configuration (

10\times 10

nodes,

6\times 6

RBF centres, RBF width factor 0.3).

Dimensionality Reduction

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

See all 145 papers

FloDR: An invertible dimensionality reduction method based on a normalising flow
Jul 28, 2026Abdallah Baraka, Daniel ProbstData Visualization Visual Analytics
Bioinformatics Group Wageningen University & Research Wageningen 6708PB, The Netherlands
ScaleMAP: Preserving Local Density and Neighborhood Structure in Low-Dimensional Embeddings
May 28, 2026Rajas Poorna, Marcus T. CiceroneData Visualization Manifold Learning
Georgia Institute of Technology · School of Chemical and Biomolecular Engineering Georgia Institute of Technology · School of Chemistry and Biochemistry Georgia Institute of Technology
Normalizing Flow Maps
Oct 15, 2026Francesco M. Ruscio, Timothy James Hitge, Fred Zhangzhi Peng +6Normalizing Flows Flow Map Learning
AITHYRA · Imperial College London · TU Wien +5

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

FloDR: An invertible dimensionality reduction method based on a normalising flow

Abdallah Baraka, Daniel Probst

Bioinformatics Group Wageningen University & Research Wageningen 6708PB, The Netherlands

Data Visualization Visual Analytics Dimensionality Reduction

Open paper AskPDF arXiv

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.

Abstract

Dimensionality Reduction

Explore similar work

FloDR: An invertible dimensionality reduction method based on a normalising flow

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

Normalizing Flow Maps