Euclidean Embedding of Data Using Local Distances

Dimitris Arabadjis

arxiv: 2605.19243 · v1 · pith:CSRYQTPBnew · submitted 2026-05-19 · 💻 cs.LG · cs.AI· cs.CG

Euclidean Embedding of Data Using Local Distances

Dimitris Arabadjis This is my paper

Pith reviewed 2026-05-20 07:41 UTC · model grok-4.3

classification 💻 cs.LG cs.AIcs.CG

keywords euclidean embeddinglocal distancesneighborhood graphvariational problemeuler-lagrange equationsisometric embeddingmanifold learninggraph-based embedding

0 comments

The pith

A variational problem on a local distance graph recovers a globally consistent Euclidean embedding without prior vector data.

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

The paper sets out to recover a Euclidean embedding of data points that respects both local distances and global consistency, starting only from a weighted neighborhood graph. It formulates the task as a variational minimization in which the Euclidean distances induced by the gradients of the embedding functions are matched to the supplied on-graph distances. The resulting Euler-Lagrange equations are written in coordinate-free form so that every operator can be evaluated directly from the graph; although the equations are nonlinear, they are solved by an iterative sequence of sparse linear systems. If the method works, it supplies an isometric embedding that preserves neighboring relations and local metric structure on both synthetic manifolds and real data sets, all without any initial feature vectors or global coordinates.

Core claim

The optimal Euclidean embedding is the stationary point of a functional that equates supplied local distances to the metric induced by the differentials of the embedding coordinate functions; the associated Euler-Lagrange equations, derived without reference to any ambient coordinate system, admit an iterative numerical solution as a sequence of sparse linear problems that use only local graph operations.

What carries the argument

The variational functional that aligns on-graph distances with the Euclidean metric induced by the differentials of the embedding functions, together with its coordinate-free Euler-Lagrange equations solved iteratively as sparse linear systems.

If this is right

Local metric structure and neighboring relations are preserved in the final embedding.
The procedure approximates a global isometric embedding on both synthetic manifolds and real data sets.
All computations are performed with local graph operations and require no initial feature vectors.
The formulation remains representation-free, depending only on the weighted neighborhood distance graph.

Where Pith is reading between the lines

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

The iterative linear solver may scale to larger graphs if the sparsity pattern remains bounded.
Adding a regularization term to the variational functional could improve robustness when the input graph contains noise or missing edges.
The same local-matching principle might be applied to other target geometries beyond Euclidean space by changing the induced metric in the functional.

Load-bearing premise

A globally consistent Euclidean embedding that preserves the given local distances exists and the supplied neighborhood graph accurately reflects the underlying metric without substantial noise or missing edges.

What would settle it

On a synthetic manifold whose true Euclidean distances are known, such as a flat torus or sphere sampled densely, measure whether the pairwise distances in the recovered embedding match the ground-truth distances up to rigid motion while also preserving the original neighborhood relations on held-out edges.

Figures

Figures reproduced from arXiv: 2605.19243 by Dimitris Arabadjis.

**Figure 2.** Figure 2: Recovery of the parameterization of a 5-dimensional manifold embedded in 10-dimensional [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗

**Figure 3.** Figure 3: MNIST dataset 10-fold experiment. The measures’ box plots depict the maximum/minimum [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗

**Figure 4.** Figure 4: Reconstruction results on the Klein’s bottle. Reconstruction of the original bottle figure [PITH_FULL_IMAGE:figures/full_fig_p019_4.png] view at source ↗

**Figure 5.** Figure 5: Embedding of the flat torus in R 3 . Reconstruction of this short embedding from the neighborhoods’ graph of constant intrinsic dimension 2, across all neighborhoods. The coloring depicts the data points’ polar angle in the original figure. The (u, v) - parameterization of the manifold is used as ground truth for evaluating the embeddings’ capability to recover the manifold’s product structure. The global … view at source ↗

**Figure 6.** Figure 6: Visualization of the RNA-seq emedding. 3D projection of the computed 30-dimensional [PITH_FULL_IMAGE:figures/full_fig_p021_6.png] view at source ↗

**Figure 7.** Figure 7: FMNIST dataset 10-fold experiment. The measures’ box plots depict the maxi [PITH_FULL_IMAGE:figures/full_fig_p022_7.png] view at source ↗

read the original abstract

We study the problem of recovering a globally consistent Euclidean embedding of data, given only a local distance graph and propose a method that optimally represents these distances. The method operates solely on a neighborhood graph weighted by pairwise distances, without requiring any prior vector representation of the data. The embedding is obtained by solving a variational problem that matches local, on-graph distances to the Euclidean metric, induced by the differentials of the embedding functions. The resulting Euler-Lagrange equations are derived in a coordinate-free form, enabling direct evaluation of all operators from the distance graph alone. Though non-linear and missing an explicit expression for their non-linearity, these equations are shown to be resolved as an iteratively updated sparse linear problem. The main contributions of the proposed approach are (a) the derivation of the functional equations governing the optimal Euclidean embedding in the continuum, (b) a representation-free formulation that requires only a neighborhood distance graph and no feature vectors and (c) an estimation procedure based exclusively on local graph operations. We experimentally evaluate the resulting non-parametric algorithm on synthetic manifolds and real datasets, demonstrating consistent preservation of local metric structure and neighboring relations, while approximating the global isometric embedding.

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.

Desk Editor's Note private letter to a colleague

The paper gives a coordinate-free variational route to Euclidean embedding from a local distance graph that reduces to iterative sparse linear solves, but the iteration lacks any shown convergence to the claimed minimizer.

read the letter

The main thing to know is that the authors set up a variational problem whose Euler-Lagrange equations match local graph distances to the induced Euclidean metric, then reduce those equations to an iteratively updated sparse linear system that can be run directly on the neighborhood graph without any initial feature vectors. That combination is the actual novelty. The coordinate-free derivation and the claim that all operators come from the distance graph alone are cleaner than the usual Isomap or MDS pipelines, and the representation-free angle is a genuine practical advantage for settings where only pairwise local distances are available. The experiments on synthetic manifolds and real data are said to show decent local metric preservation, which at least suggests the method is not obviously broken on the examples they tried. The soft spot is exactly where the stress-test note points: the equations are acknowledged to be non-linear, yet the abstract gives no explicit form for the non-linearity and supplies no convergence argument, contraction mapping, or numerical evidence that the fixed-point iteration actually reaches a critical point of the original functional rather than an artifact. On irregular or noisy graphs this could produce embeddings that satisfy the local conditions only approximately while drifting on global consistency. The central argument therefore rests on an unverified numerical step. This is for readers already working on graph-based manifold learning or non-parametric dimensionality reduction who are looking for alternatives that stay purely on the distance graph. Someone who wants to implement or extend the method would get value from the formulation, but only after checking the full derivations and the iteration behavior. The paper is coherent enough on its own terms and the claims are specific enough to be checked that it deserves a serious referee rather than a desk reject; the referees should be asked to focus on the convergence question and the experimental controls.

Referee Report

2 major / 2 minor

Summary. The paper proposes recovering a globally consistent Euclidean embedding of data from a local distance graph alone, without prior vector representations. It formulates this as a variational problem matching on-graph local distances to the Euclidean metric induced by the differentials of the embedding functions, derives the Euler-Lagrange equations in coordinate-free form directly from the distance graph, and resolves the resulting non-linear system via iteratively updated sparse linear solves. Experiments on synthetic manifolds and real datasets are reported to demonstrate local metric preservation and approximation of global isometry.

Significance. If the variational derivation and iterative solver are shown to converge to a true minimizer of the distance-matching functional, the approach would provide a representation-free, graph-only method for isometric manifold embedding. This could strengthen non-parametric dimensionality reduction techniques by avoiding explicit feature vectors and relying solely on local operations, with potential value in settings where only neighborhood distances are available.

major comments (2)

[§3] §3 (Variational formulation and EL derivation): the coordinate-free derivation of the Euler-Lagrange equations from the distance-matching functional is stated without explicit intermediate steps, variation computation, or verification that the resulting system indeed corresponds to a critical point of the original energy; the non-linearity is asserted but never written explicitly, preventing direct assessment of the linearization.
[§4.2] §4.2 (Iterative linear solver): the claim that the non-linear EL system is resolved by iteratively updated sparse linear problems lacks any convergence analysis, fixed-point contraction argument, or numerical evidence that the iteration reaches a critical point of the original variational functional rather than an artifact of the linearization; this is load-bearing for the optimality and global-consistency claims, especially on irregular or noisy graphs.

minor comments (2)

[Abstract] The abstract and introduction should explicitly state the precise form of the variational functional being minimized.
[Experiments] Figure captions and experimental tables would benefit from clearer indication of which metrics quantify global isometry versus local distance preservation.

Simulated Author's Rebuttal

2 responses · 1 unresolved

We thank the referee for the constructive and detailed comments. We address each major comment point by point below, indicating the revisions we will make to strengthen the manuscript.

read point-by-point responses

Referee: [§3] §3 (Variational formulation and EL derivation): the coordinate-free derivation of the Euler-Lagrange equations from the distance-matching functional is stated without explicit intermediate steps, variation computation, or verification that the resulting system indeed corresponds to a critical point of the original energy; the non-linearity is asserted but never written explicitly, preventing direct assessment of the linearization.

Authors: We agree that the derivation in §3 is presented in a condensed manner. In the revised manuscript we will expand this section to include the explicit first variation of the distance-matching functional, all intermediate steps in the coordinate-free derivation of the Euler-Lagrange equations, a verification that the resulting system is a critical point of the original energy, and an explicit statement of the non-linear terms to permit direct assessment of the linearization. revision: yes
Referee: [§4.2] §4.2 (Iterative linear solver): the claim that the non-linear EL system is resolved by iteratively updated sparse linear problems lacks any convergence analysis, fixed-point contraction argument, or numerical evidence that the iteration reaches a critical point of the original variational functional rather than an artifact of the linearization; this is load-bearing for the optimality and global-consistency claims, especially on irregular or noisy graphs.

Authors: We acknowledge that the current manuscript does not contain a formal convergence analysis or fixed-point argument. While the reported experiments demonstrate practical performance, we agree that additional targeted numerical evidence is required. In the revision we will add experiments that track the variational energy across iterations and explicitly demonstrate convergence behavior on irregular and noisy graphs. A complete theoretical convergence guarantee, however, lies beyond the scope of the present work. revision: partial

standing simulated objections not resolved

A rigorous theoretical convergence analysis or contraction-mapping argument establishing that the iteration reaches a critical point of the original variational functional on arbitrary graphs.

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper defines a variational functional whose minimizer is required to match supplied local graph distances to the Euclidean metric induced by the embedding differentials. The Euler-Lagrange equations are then obtained from this functional via standard coordinate-free calculus of variations, after which an iterative sparse linear solver is introduced as a numerical scheme. No quoted step reduces the final embedding or the governing equations to a fitted parameter, a self-referential definition, or a load-bearing self-citation; the entire chain begins from the explicitly stated distance-matching objective and proceeds by direct derivation and discretization. The method is therefore self-contained against its own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the existence of an isometric Euclidean embedding and on the neighborhood graph faithfully encoding local metric structure; no free parameters or invented entities are explicitly introduced in the abstract.

axioms (2)

domain assumption A globally consistent Euclidean embedding exists that preserves the supplied local distances.
Invoked to guarantee that the variational problem has a meaningful solution.
domain assumption The neighborhood graph accurately reflects the underlying local metric.
Required for the local-distance matching to produce a globally consistent embedding.

pith-pipeline@v0.9.0 · 5727 in / 1142 out tokens · 50672 ms · 2026-05-20T07:41:28.176239+00:00 · methodology

discussion (0)

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 unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

The embedding is obtained by solving a variational problem that matches local, on-graph distances to the Euclidean metric induced by the differentials of the embedding functions; the resulting Euler-Lagrange equations are resolved as an iteratively updated sparse linear problem.
IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

div(dφ̃) = div(αQ̃) … Q̃ = Rφ̃ Lφ̃ᵀ … svd(Bφ̃)

What do these tags mean?

matches: 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.

Reference graph

Works this paper leans on

23 extracted references · 23 canonical work pages

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Petar Veliˇckovi´c, Guillem Cucurull, Arantxa Casanova, Adriana Romero, Pietro Liò, and Yoshua Bengio. Graph attention networks. InInternational Conference on Learning Representations, 2018. A Inner product of differential 1-forms and their divergence Concerning the notation adopted for representing vectors, matrices and tensors and their contractions, Ei...

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original

+O(ϵ Q 2). As a consequence, theQ- variation ofJ[φ, Q]read d dϵQ J[φ, Q] ϵQ=0 =− Z Ω αiδQi j ∧⋆ε j (39) while the φ - variations read d dϵφ J[φ, Q] ϵφ=0 = R Ω dδφj ∧⋆ε j. Then by the generalized Stokes theorem, applied tod(δφ j ∧⋆ε j) =dδφ j ∧⋆ε j +δφ jd∧⋆ε j, theφ- variations ofJ[φ, Q]read d dϵφ J[φ, Q] ϵφ=0 = Z ∂Ω ⋆εjδφj − Z Ω δφjd ⋆ εj = Z Ω δφjdiv(εj)...

work page

[1] [1]

J. B. Tenenbaum, V . de Silva, and J. C. Langford. A global geometric framework for nonlinear dimension- ality reduction.Science, 290(5500):2319–2323, 2000

work page 2000

[2] [2]

W. S. Torgerson.Theory and Methods of Scaling. John Wiley and Sons, New York, 1958

work page 1958

[3] [3]

Minimum-distortion embedding.Foundations and Trends in Machine Learning, 14(3):211–378, 2021

Akshay Agrawal, Alnur Ali, and Stephen Boyd. Minimum-distortion embedding.Foundations and Trends in Machine Learning, 14(3):211–378, 2021

work page 2021

[4] [4]

S. T. Roweis and L. K. Saul. Nonlinear dimensionality reduction by locally linear embedding.Science, 290(5500):2323–2326, 2000

work page 2000

[5] [5]

L. K. Saul and S. T. Roweis. Think globally, fit locally: Unsupervised learning of low dimensional manifolds.Journal of Machine Learning Research, 4:119–155, 2003

work page 2003

[6] [6]

Principal manifolds and nonlinear dimensionality reduction via tangent space alignment.SIAM Journal on Scientific Computing, 26(1):313–338, 2004

Zhang Zhenyue and Zha Hongyuan. Principal manifolds and nonlinear dimensionality reduction via tangent space alignment.SIAM Journal on Scientific Computing, 26(1):313–338, 2004

work page 2004

[7] [7]

Belkin and P

M. Belkin and P. Niyogi. Laplacian eigenmaps for dimensionality reduction and data representation. Neural Computation, 15(6):1373–1396, 2003

work page 2003

[8] [8]

Donoho and Carrie Grimes

David L. Donoho and Carrie Grimes. Hessian eigenmaps: Locally linear embedding techniques for high-dimensional data.Proceedings of the National Academy of Sciences, 100(10):5591–5596, 2003

work page 2003

[9] [9]

van der Maaten and G

L. van der Maaten and G. Hinton. Visualizing data using t-sne.Journal of Machine Learning Research, 9: 2576–2605, 2008

work page 2008

[10] [10]

Perozzi, R

B. Perozzi, R. Al-Rfou, and S. Skiena. Visualizing large-scale and high-dimensional data. InWWW ’16: 25th International Conference on World Wide Web, pages 287–297, 2016. 12

work page 2016

[11] [11]

McInnes, J

L. McInnes, J. Healy, N. Saul, and J. Melville. Umap: Uniform manifold approximation and projection. Journal of Open Source Software, 29(3):861, 2018

work page 2018

[12] [12]

Griffiths and J

P. Griffiths and J. Harris.Principles of Algebraic Geometry. John Wiley and Sons, New York, 1978

work page 1978

[13] [13]

Estimating the intrinsic dimension of datasets by a minimal neighborhood information.Scientific Reports, 7, 2017

Elena Facc, Maria d’Errico, Alex Rodriguez, and Alessandro Laio. Estimating the intrinsic dimension of datasets by a minimal neighborhood information.Scientific Reports, 7, 2017

work page 2017

[14] [14]

Postma L.J.P

E.O. Postma L.J.P. van der Maaten and H.J. van den Herik. Dimensionality reduction: A comparative review. Technical Report TiCC-TR 2009-005, Tilburg University, 2009

work page 2009

[15] [15]

Uniform manifold approximation and projection (umap), 2025

Connor Meehan, Jonathan Ebrahimianand Stephen Meehanand, and Wayne Moore. Uniform manifold approximation and projection (umap), 2025. URL https://www.mathworks.com/matlabcentral/ fileexchange/71902-uniform-manifold-approximation-and-projection-umap

work page 2025

[16] [16]

R. R. Coifman and S. Lafon. Diffusion maps.Applied and Computational Harmonic Analysis, 21(1):5–30, 2006

work page 2006

[17] [17]

Gdc data portal, 2025

National Cancer Institute. Gdc data portal, 2025. URL https://portal.gdc.cancer.gov/. Accessed: 2025-12-17

work page 2025

[18] [18]

G. E. Hinton and R. R. Salakhutdinov. Reducing the dimensionality of data with neural networks.Science, 313(5786):504–507, 2006

work page 2006

[19] [19]

Auto-encoding variational bayes

Diederik P Kingma and MaxWelling. Auto-encoding variational bayes. InInternational Conference on Learning Representations (ICLR), 2014

work page 2014

[20] [20]

Neural operator: learning maps between function spaces with applications to pdes.Journal of Machine Learning Research, 24(1), 2023

Nikola Kovachki, Zongyi Li, Burigede Liu, Kamyar Azizzadenesheli, Kaushik Bhattacharya, Andrew Stuart, and Anima Anandkumar. Neural operator: learning maps between function spaces with applications to pdes.Journal of Machine Learning Research, 24(1), 2023

work page 2023

[21] [21]

Gomez, Łukasz Kaiser, and Illia Polosukhin

Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N. Gomez, Łukasz Kaiser, and Illia Polosukhin. Attention is all you need. InProceedings of the 31st International Conference on Neural Information Processing Systems, page 6000–6010, 2017

work page 2017

[22] [22]

Graph attention networks

Petar Veliˇckovi´c, Guillem Cucurull, Arantxa Casanova, Adriana Romero, Pietro Liò, and Yoshua Bengio. Graph attention networks. InInternational Conference on Learning Representations, 2018. A Inner product of differential 1-forms and their divergence Concerning the notation adopted for representing vectors, matrices and tensors and their contractions, Ei...

work page 2018

[23] [23]

original

+O(ϵ Q 2). As a consequence, theQ- variation ofJ[φ, Q]read d dϵQ J[φ, Q] ϵQ=0 =− Z Ω αiδQi j ∧⋆ε j (39) while the φ - variations read d dϵφ J[φ, Q] ϵφ=0 = R Ω dδφj ∧⋆ε j. Then by the generalized Stokes theorem, applied tod(δφ j ∧⋆ε j) =dδφ j ∧⋆ε j +δφ jd∧⋆ε j, theφ- variations ofJ[φ, Q]read d dϵφ J[φ, Q] ϵφ=0 = Z ∂Ω ⋆εjδφj − Z Ω δφjd ⋆ εj = Z Ω δφjdiv(εj)...

work page