Learning Locally Adaptive Metrics that Enhance Structural Representation with $\texttt{LAMINAR}$

Christian Kleiber; Tobias Buck; William H. Oliver

REVIEW 4 major objections 6 minor 15 references

Learning Locally Adaptive Metrics that Enhance Structural Representation with $\texttt{LAMINAR}$

T0 review · 4 major / 6 minor · reviewed 2026-08-12 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read The paper presents LAMINAR, an unsupervised method that learns a locally adaptive distance metric from data by transporting the data to a uniform sphere with a continuous normalizing flow and reading off a Riemannian metric from the…

desk verdict LAMINAR's pipeline is a genuine combination of normalizing flows and Jacobian metrics, but the paper's center rests on an unproven locality-preservation assumption and the validation is too qualitative to carry that weight. read the letter →

arxiv 2411.08557 v1 pith:A7YMSGAU submitted 2024-11-13 cs.LG

Christian Kleiber , William H. Oliver , Tobias Buck This is my paper · ORCID

classification cs.LG

keywords locallyadaptivemetricnormalizingflowsinversetransformsamplingdensity-baseddistanceRiemannianlearningk-medoidsclustering

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

LAMINAR is an unsupervised method that learns a distance metric directly from the shape of a data set, without requiring the user to choose a metric in advance. It transports the data to a uniform distribution inside a unit sphere using a continuous normalizing flow, then uses the Jacobian of that transformation to define a locally varying Riemannian metric tensor at each point. Distances between nearby points are Mahalanobis distances with the averaged local tensor, and longer distances are shortest paths through a $k$-nearest-neighbour graph whose edges carry those local weights. The paper's claim is that these distances are structurally informative: points in the same dense mode end up closer and points in separate modes farther apart than under the Euclidean metric, which would make downstream distance-based analysis more faithful to the data's geometry.

What carries the argument

The machinery is a continuous planar flow (a neural ODE) that maps the input distribution to a standard normal, followed by a radial rescaling that converts it into a uniform distribution inside the $d$-dimensional unit sphere. This combined map acts as a pseudo-cdf: its output is uniform by construction, and the paper assumes it preserves locality. The Jacobian of the combined map, $J_{\text{total}} = J_{\text{to sphere}} J_{\text{flow}}$, defines a Riemannian metric tensor $\Sigma = (J_{\text{total}}^T J_{\text{total}})^{-1}$ at each point. Neighbourhood structure is obtained from a Euclidean $k$-nearest-neighbour graph on the transformed points, edge weights are Mahalanobis distances with the averaged tensor, and global distances are computed as Dijkstra shortest paths on this graph.

What would settle it

Construct a data set with two high-density clusters connected by a thin low-density bridge, where the Euclidean metric would cut across the gap. If LAMINAR's locality-preservation claim is correct, its shortest paths should stay on the bridge and the ratio of between-cluster to within-cluster distances should be larger than the Euclidean ratio; if instead the learned graph connects points across the gap or the distance ratios do not move in that direction, the central claim is falsified. A quantitative check would compare LAMINAR distances against a kernel-density-estimate geodesic on the same data.

Watch

Extended reading notes

Core claim

The central claim is that a density-based locally adaptive metric can be learned without a pre-specified metric by using a normalizing flow as a pseudo-cdf. After training, the flow maps the data to a uniform ball, and because the transformation is invertible and locally smooth, Euclidean neighborhoods in the transformed space stand in for structurally meaningful neighborhoods in the original space. LAMINAR defines the metric tensor at each point as $\Sigma = (J_{\text{total}}^T J_{\text{total}})^{-1}$, where $J_{\text{total}}$ is the Jacobian of the full transformation, and sets the distance between neighbouring points to the Mahalanobis distance using the averaged tensors of the two points. The paper demonstrates on toy data that the learned tensors reproduce analytically known ground-truth metrics, that distances travel along dense modes, and that $k$-medoids clustering with LAMINAR distances improves Jaccard similarity over the Euclidean metric for almost all tested cases.

Load-bearing premise

The load-bearing premise is that the learned transformation to the uniform sphere preserves locality, so that Euclidean nearest-neighbour searches in the transformed space identify the same structurally meaningful neighbourhoods as in the original data; the paper asserts this without proof.

Editorial extensions

If this is right

Distances computed with LAMINAR make movement along dense modes cheaper and movement across sparse regions more expensive than Euclidean distance does.
Any point-based downstream task that uses distances—clustering, nearest-neighbour classification, visualization—can adopt LAMINAR distances without changing the algorithm.
Because the metric is learned from the data, users do not need to hand-specify a metric or a separate density estimator, removing a common manual step in density-based distance methods.
In the paper's $k$-medoids experiments, Jaccard similarity between recovered and ground-truth clusters is higher with LAMINAR than with Euclidean distances for almost all data sets tested.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Beyond the paper: if the locality-preservation assumption survives a careful test, the sphere-transformation step could become a generic preprocessing layer for any metric-based algorithm, not just $k$-medoids.
Beyond the paper: because infinitely many flows produce the same uniform sphere, different flow architectures will produce different locally adaptive metrics; the choice of flow is thus likely to matter for downstream performance, and the paper's own suggestion of an optimal-transport-based flow points to a concrete comparative experiment.
Beyond the paper: extending the metric to out-of-distribution points by inverting the flow would let LAMINAR serve regression and classification tasks, since the current proof-of-concept can only measure distances between existing data points.

Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The core idea in LAMINAR is worth taking seriously. Training a continuous normalizing flow to map data to a uniform sphere and then using the flow Jacobian to define a local Mahalanobis metric is a clean way to make distances data-dependent without requiring a user-specified metric. The specific pipeline—continuous flow, radial transform to the sphere, kNN graph in the pseudo-cdf, shortest paths with Jacobian-based edge weights—is not verbatim in the cited literature. The paper is clearly written, the code is public, and the authors honestly flag the non-uniqueness issue (rotations of the flow give the same density).

The weak point is exactly what the stress-test note identifies. Section 2 asserts that the pseudo-cdf transformation 'behaves like a cumulative distribution function ... in the sense that it is uniform and that locality is preserved,' but this is not proved. A likelihood-trained continuous flow can compress high-density regions and stretch low-density ones; nothing in the loss controls the Jacobian's singular values or a bi-Lipschitz constant. If Euclidean neighborhoods in the uniform sphere do not correspond to structural neighborhoods in the original data, the kNN edges—and therefore the shortest-path distances—are built on the wrong pairings, even if the local metric tensor is correctly derived. The paper provides no quantitative test that isolates graph topology, and the qualitative comparison in Figure 2 does not resolve this.

The validation is otherwise thin. The k-medoids clustering comparison in Section 5 shows Jaccard improvements over Euclidean distances, but it does not compare against the closest prior methods: Sorrenson et al.'s flow-based distance or Sharma and Johnston's entropy-based method. The toy examples are useful for intuition but not sufficient to establish that the improvement is robust.

These soft spots are significant but not disqualifying. The paper is an honest proof-of-concept, and the underlying construction is coherent. To make it publishable, the authors need to provide either a theoretical guarantee or strong empirical evidence that the pseudo-cdf preserves local neighborhoods, and they need a quantitative benchmark against existing density-based metric methods. If those are supplied, the contribution is real.

My recommendation: send it to peer review, but with the clear expectation of a major revision. The idea deserves referee time; the current validation does not yet earn acceptance.

Referee Report

4 major / 6 minor

Summary. LAMINAR is an unsupervised algorithm that learns a locally adaptive Riemannian metric on d-dimensional point data. It trains a continuous planar normalizing flow (Eq. 2) to map the data to a standard normal, then applies the radial inverse-CDF transform of Eq. (3) so that the images fill the unit ball uniformly, in what the authors call a pseudo-cdf sense. A Euclidean k-nearest-neighbour graph is built in this transformed space; each edge is weighted by the Mahalanobis distance of Eq. (4) using the Jacobian-derived tensor Sigma = (J^T J)^(-1) of Eq. (5); and pairwise distances are computed as Dijkstra shortest paths. Validation is threefold: comparison of learned metric tensors with ground-truth pullback metrics for four transformations of a uniform circle (Figs. 1-2), qualitative distance-field comparisons (Figs. 4-5), and a k-medoids clustering benchmark (Fig. 6) in which LAMINAR distances improve Jaccard scores over Euclidean distances in 'almost all' cases. The paper is explicitly a proof-of-concept for physical-science applications, and the code is released on GitHub.

Significance. If the results hold, LAMINAR offers a modular, unsupervised way to derive density-adaptive distances for point clouds, avoiding an explicit user-specified metric on the data space; because the continuous planar flow is a diffeomorphism and the local tensor is read off its Jacobian, the construction is conceptually clean and the local metric is machine-checkable (PyTorch autograd for J_flow, analytic J_to_sphere). The synthetic ground-truth tests in Fig. 2 are a genuine strength: the target tensors come from known transformations rather than fitted parameters, so the validation is not circular, and the authors explicitly acknowledge the proof-of-concept status, the restriction to observed points, and the boundary effects. The released code and the honest limitation statements are further strengths. The potential value, if the empirical support is completed, is a simple alternative to recent flow-based metric learning (notably Sorrenson et al. [12]) for physical-science clustering; the missing part is quantitative evidence that the graph topology produced by the pseudo-cdf kNN search is correct, which is what the major comments ask for.

major comments (4)

[§2, Eqs. (2)–(5)] The load-bearing assertion that the pseudo-cdf transformation 'preserves locality' is stated in §2 without proof, and the entire kNN graph — and therefore every global distance from Dijkstra's algorithm — depends on it. The continuous planar flow of Eq. (2) is trained by maximum likelihood only, and nothing in that objective controls the conditioning of J_flow or the curvature of the map, so in the finite-sample regime the Euclidean kNN neighborhoods in the unit ball need not coincide with the pullback-metric neighborhoods in the data space; the radial transform of Eq. (3) cannot undo an arbitrarily anisotropic flow Jacobian. Because the continuous planar flow with Lipschitz h is in fact a diffeomorphism, the locality assertion is plausibly correct in the asymptotic limit, but the paper should (i) state this property together with the needed regularity conditions, and (ii) validate the graph topology directly in the Fig. 2 setting, for instance by reporting the fraction of kNN edges in the pseudo-cdf that connect points that are local neighbors under the ground-truth transformation, as a function of k and sample size. Without such a test, the clustering improvements in §5 cannot be attributed to correct graph topology. Relatedly, the edge weight of Eq. (4) uses a one-point quadrature (the average of the endpoint tensors) for what should be a line integral of the metric; the error of that approximation is controlled by second derivatives of the flow, which are neither bounded nor discussed.
[§3] The explanatory argument for the discrepancy in Fig. 2 is mathematically incorrect. The paper states that one should not expect a perfect match because the flow could learn 'T(x) vs. T(Rx)' for a rotation R; however, if R is applied in the transformed space, the target distribution on the ball is unchanged and both the metric tensor Sigma of Eq. (5) and the Euclidean kNN graph are invariant under orthogonal post-composition, so T and R∘T yield identical LAMINAR distances. The real sources of deviation are the non-identifiability of the flow beyond rotations, training non-convergence, and boundary effects, and these should be discussed instead. In addition, the Wasserstein discrepancies between ground-truth and LAMINAR tensors are computed for Fig. 2 but never reported numerically, so the claim that 'LAMINAR is able to learn the correct metric' rests on visual inspection of colour wheels; please report the mean and per-transformation discrepancies.
[§2 and §5] The manuscript omits the hyperparameters needed to reproduce the results. The value of k in the §2 kNN construction is never stated (k=25 in §4 refers only to the out-of-distribution contour estimator), and the flow architecture (number of layers, hidden width, activation, optimizer, learning rate, number of training steps, ODE solver tolerances) and the toy-data sample sizes are not given. Because a k that is too small disconnects the graph (leaving Dijkstra distances undefined or infinite) and a k that is too large can short-circuit across density modes, the choice of k is load-bearing for the method; the paper should report it, provide a sensitivity analysis, and state how it was chosen. Without these details the results in Figs. 2 and 6 are not reproducible.
[§5] The downstream evaluation compares LAMINAR only with the Euclidean metric, yet §1 positions the work as improving on existing density-based and flow-based metric-learning implementations, in particular Sorrenson et al. [12], which is credited with practical convergence. To substantiate the claim that LAMINAR 'enhances structural representation,' the clustering benchmark should include at least one density-based baseline (for example Sajama and Orlitsky [9] or [12]); the current comparison only shows that the adaptive metric helps on the specific toy geometries of Fig. 6. Furthermore, the Jaccard values in Fig. 6 appear to come from single runs, with no error bars, no stated number of data points, and no table of all cluster scores, and the phrase 'improve cluster extraction for (almost) all cases' is not backed by a complete enumeration of the results.

minor comments (6)

[§2, Eq. (3)] The text calls F(·) 'the CDF of the multivariate Gaussian'; the inner term 1 − Γ(d/2, r²/2)/Γ(d/2) is the survival function of the chi (radius) distribution of ||z||, so F is the d-th root of the chi CDF; please correct the terminology.
[§3 and Fig. 1] The four transformations in Fig. 1 are not named in the text or the figure captions; only the linear shear is identified in the body of §3, so the reader cannot tell which panel corresponds to which transformation.
[§5] The paper does not state how the number of medoids was chosen for Fig. 6; if it was set to the number of ground-truth clusters, this is a form of supervision that should be acknowledged, and the 'unsupervised' framing should be qualified accordingly.
[§2] Minor typos and infelicities: 'Euclideank-nearest-neighbour' is missing a space, 'Dijkstras's algorithm' has a misplaced apostrophe, and the sentence describing the ODE as a 'continuous transformation from any d-dimensional initial distribution to a d-dimensional standard-normal distribution' overstates what likelihood training guarantees (it holds only within the model family and on the training distribution).
[§3 and §5] The statement that 'It is difficult to quantitatively assess LAMINAR's performance' is in tension with the quantitative Jaccard-based assessment in §5; clarifying that clustering quality serves as a proxy for structural meaningfulness would resolve the apparent tension.
[§4] The claim that 'it is clear that LAMINAR is able to learn and emphasise the structure implicit to the data sets' overstates what qualitative colour maps can establish; a quantitative summary (for example, distance-ratio statistics or summary values for the fields in Fig. 5) would be needed to support the claim.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: LAMINAR's metric is a pullback construction trained by likelihood and checked against external ground-truth transformations.

full rationale

I walked the derivation chain. The metric tensor in Eq. 5 is defined as the inverse pullback of the Euclidean metric under the learned normalizing flow; this is a construction, not a fitted prediction. The flow is trained solely by maximum likelihood (Eq. 2), and no parameter is fitted to a sub-segment of data to force the downstream distances or Jaccard scores. Section 3 compares against analytically known ground-truth metrics from external transformations of a uniform circle, so that validation is independent of the fitted values. Section 5's k-medoids comparison is a demonstration, not a prediction of held-out constants. The paper contains no self-citation chain: references [10, 11] are not by the present authors, and [12] is cited as prior work, not as a warrant for LAMINAR's core construction. The only load-bearing assertion that is not proven is the Section 2 claim that the pseudo-cdf transformation 'preserves locality', which is needed for the kNN graph to be structurally meaningful. This is an unproved geometric assumption, not a circular reduction: the conclusion does not enter the training loss or the definition of the metric tensor. It is therefore a correctness risk, not a circularity. Consequently, no step meets the evidentiary bar for circularity.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

The method relies on standard normalizing flow mathematics and on the domain-specific assumption that the flow's Jacobian defines a useful metric. No new physical entities are introduced. The main unproven premise is the locality-preservation claim.

free parameters (2)

k (number of nearest neighbors in pseudo-cdf)
Not specified in the paper; controls the adjacency graph and thus the metric paths.
Flow architecture hyperparameters (network width, depth, learning rate)
Not reported; chosen by the authors, affect the quality of the learned flow and metric.

assumptions (4)

standard math The continuous planar flow equations (Eq. 1-2) are a valid application of the instantaneous change of variables.
Borrowed from Chen et al. [3]; standard normalizing flow theory.
standard math The radius transformation in Eq. 3 maps a multivariate Gaussian to a uniform distribution in the unit sphere.
Follows from the CDF of the chi distribution; used to define the pseudo-cdf.
ad hoc to paper The transformation to the pseudo-cdf preserves locality.
Asserted in Section 2 without proof; crucial for the correctness of the kNN graph in the transformed space.
domain assumption The Mahalanobis distance with metric tensor Sigma = (J^T J)^-1 is a meaningful density-based distance.
Inherited from Sharma and Johnston [10] and used without derivation in this paper.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Learning Locally Adaptive Metrics that Enhance Structural Representation with $\texttt{LAMINAR}$." pith.science (2026). https://pith.science/paper/A7YMSGAU

@misc{pith2026241108557,
  author       = {Pith},
  title        = {Pith review of: Learning Locally Adaptive Metrics that Enhance Structural Representation with $\textttLAMINAR$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/A7YMSGAU}},
  note         = {Machine review of arXiv:2411.08557}
}

abstract

We present $\texttt{LAMINAR}$, a novel unsupervised machine learning pipeline designed to enhance the representation of structure within data via producing a more-informative distance metric. Analysis methods in the physical sciences often rely on standard metrics to define geometric relationships in data, which may fail to capture the underlying structure of complex data sets. $\texttt{LAMINAR}$ addresses this by using a continuous-normalising-flow and inverse-transform-sampling to define a Riemannian manifold in the data space without the need for the user to specify a metric over the data a-priori. The result is a locally-adaptive-metric that produces structurally-informative density-based distances. We demonstrate the utility of $\texttt{LAMINAR}$ by comparing its output to the Euclidean metric for structured data sets.

Figures

Figures reproduced from arXiv: 2411.08557 by the authors.

**Figure 2.** Comparison of ground-truth and LAMINAR metric tensors produced using data in [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 4.** The distance distribution from a query point (red cross) on four toy data sets found with [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗

Figures from the paper (2 more)

**Figure 5.** Figure 5: A comparison of distances from query points (marked with a green cross) produced via [PITH_FULL_IMAGE:figures/full_fig_p004_5.png]

**Figure 6.** Figure 6: Comparison of the LAMINAR (top) and Euclidean (bottom) metrics in the downstream clustering tool, k-medoids. Values shown next to the clusters are Jaccard similarities between the ground-truth clusters and the best-fitting predicted cluster (colours). Even though the g…

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

15 extracted references · 13 canonical work pages

[12]

Learning Distances from Data with Normalizing Flows and Score Matching

Peter Sorrenson, Daniel Behrend-Uriarte, Christoph Schnörr, and Ullrich Köthe. Learning dis- tances from data with normalizing flows and score matching. arXiv preprint arXiv:2407.09297, 2024

work page Pith review arXiv 2024
[9]

Estimating and computing density based distance metrics

Sajama and Alon Orlitsky. Estimating and computing density based distance metrics. In Proceedings of the 22nd international conference on Machine learning , pages 760–767, 2005

work page 2005
[1]

Semi-supervised Learning with Density Based Distances

Avleen S Bijral, Nathan Ratliff, and Nathan Srebro. Semi-supervised learning with density based distances. arXiv preprint arXiv:1202.3702, 2012

work page Pith review arXiv 2012
[2]

Measure based regularization

Olivier Bousquet, Olivier Chapelle, and Matthias Hein. Measure based regularization. Advances in Neural Information Processing Systems , 16, 2003

work page 2003
[3]

Ricky T. Q. Chen, Yulia Rubanova, Jesse Bettencourt, and David Duvenaud. Neural ordinary differential equations, 2019. URL https://arxiv.org/abs/1806.07366

arXiv 2019
[4]

Exact computation of a manifold metric, via lipschitz embeddings and shortest paths on a graph

Timothy Chu, Gary L Miller, and Donald R Sheehy. Exact computation of a manifold metric, via lipschitz embeddings and shortest paths on a graph. In Proceedings of the F ourteenth Annual ACM-SIAM Symposium on Discrete Algorithms , pages 411–425. SIAM, 2020

work page 2020
[5]

A note on two problems in connexion with graphs

Edsger W Dijkstra. A note on two problems in connexion with graphs. Numerische mathematik, 1(1):269–271, 1959

1959
[6]

Nonhomogeneous euclidean first-passage percolation and distance learning

Pablo Groisman, Matthieu Jonckheere, and Facundo Sapienza. Nonhomogeneous euclidean first-passage percolation and distance learning. Bernoulli, 28(1):255–276, 2022

work page 2022

Show all 15 references

[7]

Shortest path through random points

Sung Jin Hwang, Steven B Damelin, and Alfred O Hero III. Shortest path through random points. 2016

2016
[8]

Anna Little, Daniel McKenzie, and James M. Murphy. Balancing geometry and density: Path distances on high-dimensional data. SIAM Journal on Mathematics of Data Science , 4(1):72–99,
[10]

Johnston

Sanjib Sharma and Kathryn V . Johnston. A group finding algorithm for multidimensional data sets. The Astrophysical Journal, 703(1):1061–1077, sep 2009. doi: 10.1088/0004-637x/703/1/

2009 doi
[11]

Multidimensional density estimation and phase-space structure of dark matter haloes

Sanjib Sharma and Matthias Steinmetz. Multidimensional density estimation and phase-space structure of dark matter haloes. Monthly Notices of the Royal Astronomical Society , 373(4): 1293–1307, December 2006. doi: 10.1111/j.1365-2966.2006.11043.x

2006
[13]

Fermat dis- tances: Metric approximation, spectral convergence, and clustering algorithms

Nicolás García Trillos, Anna Little, Daniel McKenzie, and James M Murphy. Fermat dis- tances: Metric approximation, spectral convergence, and clustering algorithms. arXiv preprint arXiv:2307.05750, 2023. 6

2023 arXiv
[1061]

URL https://doi.org/10.1088/0004-637x/703/1/1061

doi
[2022]

URL https://doi.org/10.1137/20M1386657

doi: 10.1137/20M1386657. URL https://doi.org/10.1137/20M1386657

doi

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [12]

Learning Distances from Data with Normalizing Flows and Score Matching

Peter Sorrenson, Daniel Behrend-Uriarte, Christoph Schnörr, and Ullrich Köthe. Learning dis- tances from data with normalizing flows and score matching. arXiv preprint arXiv:2407.09297, 2024

work page Pith review arXiv 2024

[2] [9]

Estimating and computing density based distance metrics

Sajama and Alon Orlitsky. Estimating and computing density based distance metrics. In Proceedings of the 22nd international conference on Machine learning , pages 760–767, 2005

work page 2005

[3] [1]

Semi-supervised Learning with Density Based Distances

Avleen S Bijral, Nathan Ratliff, and Nathan Srebro. Semi-supervised learning with density based distances. arXiv preprint arXiv:1202.3702, 2012

work page Pith review arXiv 2012

[4] [2]

Measure based regularization

Olivier Bousquet, Olivier Chapelle, and Matthias Hein. Measure based regularization. Advances in Neural Information Processing Systems , 16, 2003

work page 2003

[5] [3]

Ricky T. Q. Chen, Yulia Rubanova, Jesse Bettencourt, and David Duvenaud. Neural ordinary differential equations, 2019. URL https://arxiv.org/abs/1806.07366

arXiv 2019

[6] [4]

Exact computation of a manifold metric, via lipschitz embeddings and shortest paths on a graph

Timothy Chu, Gary L Miller, and Donald R Sheehy. Exact computation of a manifold metric, via lipschitz embeddings and shortest paths on a graph. In Proceedings of the F ourteenth Annual ACM-SIAM Symposium on Discrete Algorithms , pages 411–425. SIAM, 2020

work page 2020

[7] [5]

A note on two problems in connexion with graphs

Edsger W Dijkstra. A note on two problems in connexion with graphs. Numerische mathematik, 1(1):269–271, 1959

1959

[8] [6]

Nonhomogeneous euclidean first-passage percolation and distance learning

Pablo Groisman, Matthieu Jonckheere, and Facundo Sapienza. Nonhomogeneous euclidean first-passage percolation and distance learning. Bernoulli, 28(1):255–276, 2022

work page 2022