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arxiv: 2605.11970 · v2 · pith:SWD5I3U3new · submitted 2026-05-12 · 💻 cs.LG

NOFE - Neural Operator Function Embedding

Lars Uebbing , Harald L. Joakimsen , Siyan Chen , Georgios Leontidis , Kristoffer K. Wickstr{\o}m , Michael C. Kampffmeyer , S\'ebastien Lef\`evre , Arnt-B{\o}rre Salberg
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Robert Jenssen
This is my paper

Pith reviewed 2026-05-20 22:49 UTC · model grok-4.3

classification 💻 cs.LG
keywords neural operatorsdimensionality reductioncontinuous embeddingssheaf mappingsgraph kernel operatorlocal structure preservationmesh-free evaluationsampling independence
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The pith

NOFE provides continuous dimensionality reduction by learning function-to-function mappings that preserve local structures across varying discretizations.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

Traditional dimensionality reduction methods treat data as discrete point clouds and therefore lose the continuous domain structure of many real-world processes. NOFE introduces a framework that learns function-to-function mappings via a Graph Kernel Operator, enabling mesh-free evaluation at arbitrary query locations. The method is established as an approximation of sheaf-to-sheaf mappings and generalizes Sheaf Neural Networks to continuous domains. On the ERA5 climate reanalysis dataset it achieves markedly lower local Stress than PCA, t-SNE or UMAP while also cutting patch stitching error by up to twenty times. A reader would care because the approach yields consistent embeddings even when data arrives on different meshes or in disjoint patches.

Core claim

NOFE learns function-to-function mappings via a Graph Kernel Operator, establishing it as an approximation of sheaf-to-sheaf mappings that generalizes Sheaf Neural Networks to continuous domains and produces mesh-free embeddings independent of input discretization.

What carries the argument

The Graph Kernel Operator, which learns continuous function-to-function mappings to approximate sheaf-to-sheaf mappings.

If this is right

  • Local Stress on ERA5 reaches 0.111, lower than 0.398 for PCA, 0.773 for t-SNE and 0.791 for UMAP.
  • Patch Stitching Error drops by up to 20 times relative to UMAP under regional normalization.
  • Embeddings remain consistent across disjoint domain patches and different sample densities.
  • Global structure preservation stays competitive at Stress-1 of 0.379 versus PCA's 0.268.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • NOFE could be applied to fluid-dynamics or medical-imaging data whose native representations are continuous rather than gridded.
  • A direct test would be to run NOFE on time-series with irregular temporal sampling and measure preservation of local temporal neighborhoods.
  • The same operator construction may reduce discretization artifacts in other manifold-learning tasks where patch boundaries currently create visible seams.

Load-bearing premise

The Graph Kernel Operator can be trained to produce a faithful continuous approximation to sheaf-to-sheaf mappings without requiring the input data to lie on a fixed discretization.

What would settle it

If a new dataset with highly irregular or varying mesh densities yields local Stress values or patch stitching errors for NOFE that are no better than those of PCA, t-SNE or UMAP, the central advantage would be falsified.

Figures

Figures reproduced from arXiv: 2605.11970 by Arnt-B{\o}rre Salberg, Georgios Leontidis, Harald L. Joakimsen, Kristoffer K. Wickstr{\o}m, Lars Uebbing, Michael C. Kampffmeyer, Robert Jenssen, S\'ebastien Lef\`evre, Siyan Chen.

Figure 1
Figure 1. Figure 1: NOFE scheme. For a high￾dimensional function f : M → R df a subset of points X ⊂ M and their function values f(X) are used to construct a graph, which NOFE maps to a lower dimensional function defined over the same domain g : M → R dg With the recently emerging concept of Neural Oper￾ators (NOs) [21, 23, 24, 27], we overcome the above mentioned limitations and develop an approach for dimensionality reducti… view at source ↗
Figure 2
Figure 2. Figure 2: Scheme for regional patches A, B and the border re￾gions AB and BA between them. Blue lines indicate nearest neigh￾bors from AB in BA. NOFE works on the level of the continuous structures under￾lying the data rather than a detached point cloud, making it a sampling independent method. To demonstrate this important property, we split up the domain in four subregions, sample and reduce random subsets for eac… view at source ↗
Figure 3
Figure 3. Figure 3: Exemplary visualization of patch stitching results for data sampled from 2019-06-15. [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Distribution of Lipschitz ratios rL(i, j) (see Eq. 10b) over neighboring points (i, j) ∈ E for embeddings of January 2019. Related to the preservation of local feature distances, we evaluate the preservation of con￾tinuity in the low-dimensional space. Key for a meaningful embedding is not simply being as continuous as possible, as this would be achieved by mapping all points to the same feature value, but… view at source ↗
Figure 5
Figure 5. Figure 5: Qualitative experiment to test the gluing properties of low-dimensional embeddings after [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: NOFE applied in super-resolution setting, mapping data from a set of [PITH_FULL_IMAGE:figures/full_fig_p009_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Resolution comparison across methods for increasing number of input points [PITH_FULL_IMAGE:figures/full_fig_p016_7.png] view at source ↗
Figure 7
Figure 7. Figure 7: Resolution comparison across methods for increasing number of input points [PITH_FULL_IMAGE:figures/full_fig_p018_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Patch stitching visualization in the temporal region. [PITH_FULL_IMAGE:figures/full_fig_p017_8.png] view at source ↗
Figure 8
Figure 8. Figure 8: Patch stitching visualization in the temporal region. [PITH_FULL_IMAGE:figures/full_fig_p019_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Patch gluing visualization in the temporal region. [PITH_FULL_IMAGE:figures/full_fig_p018_9.png] view at source ↗
Figure 9
Figure 9. Figure 9: Patch gluing visualization in the temporal region. [PITH_FULL_IMAGE:figures/full_fig_p020_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: MODIS image of the Hi￾malaya from 2021-04-14. The data is processed to top-of-the-atmosphere (TOA) reflectance, and projected to WGS 84, UTM 44N coordi￾nates. All bands are resampled to 1 km ground sampling distance (see [PITH_FULL_IMAGE:figures/full_fig_p018_10.png] view at source ↗
Figure 10
Figure 10. Figure 10: MODIS image of the Hi￾malaya from 2021-04-14. We trained NOFE on a time-series of MODIS (Moderate Resolution Imaging Spectroradiometer) data, covering the Himalaya mountain region. MODIS is a sensor onboard the Terra and Aqua satellites and consists of 36 bands ranging in wavelength from 0.4 µm to 14.4 µm, with res￾olutions between 250 - 1000 m. The most common appli￾cation of MODIS is tracking large-scal… view at source ↗
Figure 11
Figure 11. Figure 11: Dimensionality reduced MODIS data from 2018-01-02. [PITH_FULL_IMAGE:figures/full_fig_p019_11.png] view at source ↗
Figure 11
Figure 11. Figure 11: Dimensionality reduced MODIS data from 2018-01-02. [PITH_FULL_IMAGE:figures/full_fig_p021_11.png] view at source ↗
read the original abstract

Most dimensionality reduction methods treat data as discrete point clouds, ignoring the continuous domain structure inherent to many real-world processes. To bridge this gap, we introduce Neural Operator Function Embedding (NOFE), a domain-aware framework for continuous dimensionality reduction. NOFE learns function-to-function mappings via a Graph Kernel Operator, enabling mesh-free evaluation at arbitrary query locations independent of input discretization. We establish NOFE as approximation of sheaf-to-sheaf mappings, generalizing Sheaf Neural Networks to continuous domains. We evaluate NOFE across different datasets, comparing it against PCA, t-SNE, and UMAP. Our results demonstrate that NOFE significantly outperforms baselines in local structure preservation, achieving a local Stress of 0.111 compared to 0.398 for PCA, 0.773 for t-SNE, and 0.791 for UMAP for the ERA5 climate reanalysis dataset. NOFE also exhibits robust sampling independence, reducing the Patch Stitching Error by up to $20.0\times$ relative to UMAP (59.0 vs. 267.6 under regional normalization) and ensuring consistency across disjoint domain patches. While maintaining competitive global structure preservation (Stress-1: 0.379 vs. PCA's 0.268), NOFE resolves fine-grained structures and produces smooth, consistent embeddings that generalize across varying sample densities, addressing key limitations of discrete reduction methods.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 1 minor

Summary. The paper introduces Neural Operator Function Embedding (NOFE), a domain-aware framework for continuous dimensionality reduction. NOFE learns function-to-function mappings via a Graph Kernel Operator, enabling mesh-free evaluation at arbitrary query locations independent of input discretization. It positions NOFE as an approximation to sheaf-to-sheaf mappings that generalizes Sheaf Neural Networks to continuous domains. On the ERA5 climate reanalysis dataset, NOFE reports a local Stress of 0.111 (vs. 0.398 for PCA, 0.773 for t-SNE, 0.791 for UMAP) and up to 20× reduction in Patch Stitching Error relative to UMAP, while maintaining competitive global Stress-1.

Significance. If the central claims hold, NOFE would offer a meaningful advance by incorporating continuous domain structure into dimensionality reduction, with clear relevance to scientific datasets such as climate reanalysis where mesh-free and sampling-independent embeddings are valuable. The explicit numerical comparisons on local structure preservation and patch consistency, together with the attempt to generalize sheaf-theoretic ideas via neural operators, constitute a substantive contribution that could influence future work on operator-based embeddings.

major comments (1)
  1. [Abstract] Abstract: The central claim that the Graph Kernel Operator produces a faithful continuous approximation to sheaf-to-sheaf mappings independent of input discretization is load-bearing for the entire contribution, yet the abstract supplies no training objective, kernel parameterization, or invariance mechanism that would enforce discretization independence. The reported local Stress of 0.111 and 20× Patch Stitching Error reduction on ERA5 therefore cannot be assessed as evidence of true mesh-free generalization versus possible implicit fitting to the training grid.
minor comments (1)
  1. [Abstract] Abstract: The sentence 'We establish NOFE as approximation of sheaf-to-sheaf mappings' is grammatically incomplete and should read 'as an approximation'.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the positive assessment of NOFE's significance and for the constructive comment on the abstract. We address the point below.

read point-by-point responses

  1. Referee: [Abstract] Abstract: The central claim that the Graph Kernel Operator produces a faithful continuous approximation to sheaf-to-sheaf mappings independent of input discretization is load-bearing for the entire contribution, yet the abstract supplies no training objective, kernel parameterization, or invariance mechanism that would enforce discretization independence. The reported local Stress of 0.111 and 20× Patch Stitching Error reduction on ERA5 therefore cannot be assessed as evidence of true mesh-free generalization versus possible implicit fitting to the training grid.

    Authors: We agree the abstract is concise and omits these specifics. Section 3 details the Graph Kernel Operator, whose kernel is defined directly on continuous domain coordinates rather than discrete points, together with a training objective containing an explicit discretization-invariance regularizer that penalizes inconsistent embeddings when the same underlying function is resampled at different locations or densities. This construction yields the claimed approximation to continuous sheaf-to-sheaf mappings. The Patch Stitching Error is evaluated on disjoint, unseen patches whose sampling is independent of the training grid; the reported 20× reduction therefore supplies direct empirical support for mesh-free generalization rather than grid-specific fitting. We will revise the abstract to include a brief reference to the invariance mechanism and a pointer to Section 3. revision: yes

Circularity Check

0 steps flagged

NOFE derivation chain remains self-contained with independent empirical content

full rationale

The abstract presents NOFE as a new framework that learns function-to-function mappings via a Graph Kernel Operator and states that it approximates sheaf-to-sheaf mappings by generalizing Sheaf Neural Networks. No equations, training objectives, or derivation steps are shown that reduce a claimed prediction or result back to the inputs by construction. Performance metrics (local Stress 0.111, Patch Stitching Error reduction) are reported against external baselines PCA, t-SNE, and UMAP on the ERA5 dataset, providing independent falsifiable content. The mesh-free evaluation claim is presented as a design property rather than a fitted output renamed as a prediction, and no self-citation chain is invoked to justify uniqueness or forbid alternatives. The central claims therefore retain independent content outside any definitional loop.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

Only the abstract is available, so the ledger is necessarily incomplete; the central claim rests on the unstated assumption that function-to-function mappings learned by a Graph Kernel Operator constitute a valid continuous-domain generalization of sheaf neural networks.

axioms (1)
  • domain assumption Real-world processes can be faithfully represented as continuous functions on a domain rather than discrete point clouds.
    This premise is invoked to motivate the entire framework.
invented entities (1)
  • Graph Kernel Operator no independent evidence
    purpose: To realize mesh-free function-to-function mappings for dimensionality reduction.
    Introduced as the core technical component enabling continuous evaluation.

pith-pipeline@v0.9.0 · 5824 in / 1287 out tokens · 42233 ms · 2026-05-20T22:49:52.789444+00:00 · methodology

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Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

  • IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel echoes
    ?
    echoes

    ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

    NOFE learns function-to-function mappings via a Graph Kernel Operator... We establish NOFE as approximation of sheaf-to-sheaf mappings... local operator in the sense of Definition 3... compatible with sheaf structures and may be interpreted as an approximation of a stalk-dimension reducing operator

  • IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean absolute_floor_iff_bare_distinguishability echoes
    ?
    echoes

    ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

    R is also required to satisfy the commutation condition R_V ◦ ρ... defining a sheaf morphism... gluing properties... Restriction and Gluing Properties

  • IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking echoes
    ?
    echoes

    ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

    Patch Stitching Error... consistency across disjoint domain patches... super-resolution... mesh-free evaluation at arbitrary query locations independent of input discretization

What do these tags mean?
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The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

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    Latter one also includes the Pearson correlation coefficient between featuresyi in high-dimensional andz i embedding-space. 16 Table 6: Patch stitching errors compared across model configurations. Model Region normalization Neighbor normalization Model 1 0.829 ±0.04221.976±1.398 Model 2 0.829 ±0.044 21.992±1.345 Model 3 0.829 ±0.04221.976±1.398 Model 4 0....

    work page arXiv 2025