NFMap: Topographic 2D Manifold Learning with Invertible Normalizing Flows

The gist NFMAP introduces an invertible, parametric dimensionality reduction method that combines Generative Topographic Mapping's fixed latent grid with the exact likelihood of a normalizing flow.

Method Overview

NFMAP maps data to a latent space split into a 2-dimensional navigable map and an off-manifold residual, ensuring full invertibility because the flow is bijective and penalizes volume contraction NFMAP maps data to a latent space of the same dimension, split into a 2-dimensional navigable map and an off-manifold residual The residual scale is the main control between local neighbourhood fidelity and global distance fidelity The flow is built from B affine coupling blocks where the mapping is invertible in closed form, and the latent vector is partitioned into a two-dimensional map and an off-manifold residual z = fθ(˜x) = (z∥, z⊥).

Prior Design

The prior factorizes over the two blocks, combining a mixture model over the 2D map with a Gaussian penalty on the residual For the map prior pmap, NFMAP uses a Gaussian Mixture Model (GMM) or, to address map crowding in larger datasets, a Student-t Mixture Model (SMM). Following GTM, component means are fixed on a regular grid of radius grid bounds and the mixing weights are uniform. The objective function minimizes a weighted sum involving the map prior, manifold penalty, volume term, graph loss, repulsion loss, and injectivity term.

Key Control Mechanisms

The residual scale σ⊥ acts as a control on the map with a volume argument showing why contracting z⊥ forces structure into the visible coordinates z∥. Decreasing σ⊥ therefore forces more of the data’s geometry into the two visible dimensions, improving global distance fidelity at the cost of local neighbourhood fidelity. The coupling between σ⊥ and the grid extent means that grid-based methods should not be tuned one parameter at a time.

Performance and Benchmarking

NFMAP is compared against t-SNE, UMAP, and Generative Topographic Mapping (GTM) across diverse datasets including chemical sets like BACE, BBBP, and ESOL. NFMAP's reported maps keep at least 99% of points inside grid bounds except for the S-curve (94%). The exact likelihood yields an out-of-distribution score that leads the k-NN distance baseline on every chemical dataset. Furthermore, at larger scales, a Student-t prior and a scaffold-aware kernel each improve scaffold separation on datasets like DRD2, hERG, and EGFR.

Conclusion

NFMAP is an invertible method that projects new points onto a trained map and carries any map position back to data space. The fixed grid provides a common reference canvas at no measurable cost in embedding quality. The framework is also easy to extend with Student-t prior and hierarchical chemical kernel improvements for larger datasets. The main limitations are that several hyperparameters affect how far the map spreads, and that points can fall outside the grid. The adaptive calibration of the grid extent is a natural next step.

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NFMap: Topographic 2D Manifold Learning with Invertible Normalizing Flows Héléna Alexandra Gaspar LovingScience 124 City Road, London EC1V 2NX helena@lovingscience.com 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. 1 Introduction Neighbour-embedding methods such as t-SNE [1] and UMAP [2] preserve local structure well and are the default in most pipelines. However, they are non-invertible. Their standard versions are non-parametric, but both have parametric counterparts [3, 4]. Generative topographic mapping [5] (GTM) fits a constrained Gaussian mixture whose centres lie on a fixed 2-D grid, giving a parametric, probabilistic, out-of-sample-capable map. It is the direct ancestor of this work and remains strong on global structure. Its limit is representational: the latentto-data mapping is an RBF network, so flexibility is bounded by the basis, and the mapping is not invertible. Normalizing flows [6–10] give exact likelihoods and invertible mappings but are not designed to produce a navigable low-dimensional map — their latent space has the same dimension as the data. Injective variants target density estimation on manifolds [9] rather than visualisation. Some flows put the main structure in their first few latent coordinates [11, 12], but they are built for density estimation, not 2-D visualisation. This work combines GTM’s fixed grid prior with a flow’s invertibility and exact likelihood, exposing a scalar governing how much of the data’s variance is forced into the visible map. Preprint. arXiv:2610.16750v1 [cs.LG] 12 Oct 2026

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Closest to NFMAP is FloDR [13], developed independently and concurrently 1. Both methods use a bijective normalizing flow whose leading coordinates form the 2D map while the remaining coordinates are retained as a residual, ensuring full invertibility and exact density evaluation. The two methods diverge in prior design and residual handling: NFMAP anchors the map prior to a fixed topographic grid and regulates the residual via a global scale σ⊥, whereas FloDR fits an unconstrained Gaussian mixture to the embedding after training and models the residual conditionally to derive local spread diagnostics. Furthermore, while FloDR relies on an explicit ordinal ranking loss to preserve global ordering, NFMAP obtains global distance fidelity geometrically, through residual variance contraction. NFMAP additionally offers a Student-t mixture prior for large-scale map crowding. We provide NFMAP as an open-source Python package available at https://github.com/hagax8/nfmap. Contributions: 1 The residual scale σ⊥ as a control on the map, with a volume argument showing why, under a bijection, contracting z⊥ forces structure into the visible coordinates z∥. 2 Empirical evidence that σ⊥ trades local against global structure. 3 A coupling analysis showing that σ⊥, the grid extent, the repulsion weight and the graph weight all act on one quantity. 4 The flow likelihood as an applicability-domain signal for projected data points. 5 Mechanisms to address map crowding at dataset scale: a general heavy-tailed Student-t prior over the map coordinates, and a domain-specific hierarchical chemical kernel demonstrating the framework’s extensibility to specialized data types. 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.

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2 Methods 2.1 Architecture Let x ∈ R D be an input observation. NFMAP first maps the data to a PCA-reduced representation of dimension d (˜x ∈ R d), then learns a bijection fθ: R d → R d built from B affine coupling blocks [8]. Block b partitions its input by a binary mask mb and applies z(b+1) = z(b) ⊙ expsb + tb, (sb, tb) = (1 − mb) ⊙ gθb mb ⊙ z(b), where gθb is a two-hidden-layer GELU network and sb←tanh(sb) bounds the log-scale for numerical stability. The final layer of each gθb is zero-initialised, so the flow begins as the identity.

1. Improved dimensionality reduction for chemical data by leveraging exact likelihood via normalizing flows, which provides an applicability-domain signal on chemical datasets. This allows for more reliable out-of-distribution detection compared to k-NN distance baselines, as NFMAP's exact log-likelihood leads the k-NN distance baseline on every chemical dataset.

2. Enhanced map structure preservation by controlling local/global trade-offs via the residual scale, specifically through the mechanism where shrinking σ⊥ transfers variance into the map, improving global distance fidelity at the cost of neighbourhood fidelity. This ensures that the right setting therefore depends on the data’s geometry rather than being a universal default.

3. Scalable structure preservation for large-scale chemical datasets by incorporating a Student-t prior and a scaffold-informed hierarchical kernel. This addresses map crowding by improving scaffold separation, as seen on EGFR where the combination of these two remedies reaches scaffold purity 0.642 and silhouette +0.104.

4. Increased robustness for data projection by ensuring full invertibility, as NFMAP's mapping is a normalizing flow, and is therefore fully invertible: any location on the map can be carried back to data space, unlike non-invertible methods like GTM whose mapping is an RBF network of bounded flexibility.

5. Enhanced representational power by utilizing a fixed grid prior that does not hinder performance, as Fixing the grid (Table 3, Figure 6) is statistically non-inferior to learning the mixture means for both global structure and local structure. This provides a common reference canvas at no measurable cost in embedding quality.

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.