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REVIEW 2 major objections 1 minor 48 references

Geodesic Learning via Unsupervised Decision Forests

T0 review · 2 major / 1 minor · reviewed 2026-05-25 · grok-4.3

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Pith's one-line read Unsupervised random forests recover geodesic distances on noisy high-dimensional manifolds by splitting on sparse feature combinations.

desk verdict URerF introduces a forest on sparse linear projections with BIC splits for geodesic approximation in noisy data, plus a new precision-recall evaluation, but the validation details are too thin to assess the central robustness claim. read the letter →

arxiv 1907.02844 v1 pith:ZNDA3DKP submitted 2019-07-05 stat.ML cs.IRcs.LGstat.ME

Meghana Madhyastha , Percy Li , James Browne , Veronika Strnadova-Neeley , Carey E. Priebe , Randal Burns , Joshua T. Vogelstein This is my paper · ORCID
classification stat.MLcs.IRcs.LGstat.ME
keywords geodesicdistancesunsupervisedrandomforestsmanifoldlearninghigh-dimensionalnoiseBICforGaussianmixturesconnectomedataprecision-recallcurves
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

The paper introduces URerF, an unsupervised random forest that approximates geodesic distances on manifolds embedded in high-dimensional data containing noise. It builds decision trees on low-dimensional sparse linear combinations of features rather than the full ambient space and selects splits using a fast Bayesian information criterion statistic for Gaussian mixture models. This structure allows the method to ignore noise dimensions while preserving the shortest-path geometry of the latent manifold. New geodesic precision-recall curves are proposed to measure recovery quality against the true manifold. Experiments show URerF maintains accuracy on simulated data and a real connectome dataset where Isomap, UMAP, and FLANN degrade.

What carries the argument

Unsupervised random forest (URerF) operating on low-dimensional sparse linear combinations of features, with splits selected by a fast Bayesian information criterion for Gaussian mixture models.

What would settle it

A simulation in which increasing the number of noise dimensions causes URerF geodesic estimates to degrade at the same rate as Isomap or UMAP estimates would falsify the claimed robustness.

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Extended reading notes

Core claim

URerF approximately learns geodesic distances in linear and nonlinear manifolds with noise by constructing unsupervised decision trees that partition the data using sparse linear combinations of features, with splits chosen via a fast BIC statistic for Gaussian mixtures; this yields distance estimates that remain accurate as noise dimensions increase, outperforming ambient-space methods on both simulated manifolds and a real connectome dataset.

Load-bearing premise

The latent manifold structure can be recovered from sparse linear combinations of the observed features without systematic bias introduced by the noise dimensions.

Editorial extensions

If this is right

  • URerF distance estimates remain stable as the number of irrelevant noise dimensions grows.
  • The method yields more accurate geodesic distances than Isomap, UMAP, or FLANN on a real connectome dataset.
  • Geodesic precision-recall curves provide a quantitative way to compare estimated distances against a known latent manifold.
  • The approach applies to both linear and nonlinear underlying manifolds.

Reading between the lines

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

  • The sparse-projection idea inside the forest could be adapted to improve distance preservation in other high-noise unsupervised tasks such as clustering.
  • If the fast BIC split rule generalizes, similar efficiency gains might appear in supervised random-forest variants that also need to ignore noise dimensions.
  • The method's success on connectome data suggests it may help in other biological network settings where observed features are high-dimensional but the underlying geometry is low-dimensional.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 1 minor

Summary. The paper introduces URerF, an unsupervised random forest that approximates geodesic distances on linear and nonlinear manifolds by operating on low-dimensional sparse linear feature combinations and using a fast BIC statistic for two-component GMM splits to select splits efficiently. It claims robustness to high-dimensional noise (unlike Isomap, UMAP, and FLANN) and superior performance on a real connectome dataset, supported by new geodesic precision-recall curves and experiments on simulated and real data.

Significance. If the empirical superiority holds under the stated conditions, the method could offer a practical alternative for geodesic recovery in noisy high-dimensional settings such as connectomics, where ambient-space methods degrade. The introduction of geodesic precision-recall curves is a useful evaluation contribution.

major comments (2)
  1. [§3.2] §3.2: The fast BIC approximation for GMM splits on 1-D projections is load-bearing for the central claim of noise robustness, yet the section provides no analysis or simulation demonstrating that the selected directions correlate with the latent manifold tangent (rather than variance-maximizing noise dimensions) when p ≫ n.
  2. [Experiments] Experiments section: The abstract and results claim comparative superiority on simulated and connectome data, but the manuscript supplies neither algorithmic pseudocode, statistical significance tests, error-bar details, nor ablation studies on the sparse-projection and BIC components; this prevents assessment of whether the reported gains are reproducible or attributable to the proposed mechanism.
minor comments (1)
  1. Notation for the sparse projection matrix and the resulting forest distance is introduced without an explicit equation reference, making it difficult to trace how the final distance approximates the geodesic.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments, which highlight areas where the manuscript can be strengthened. We address each major comment below and commit to revisions that improve the clarity and rigor of the work.

read point-by-point responses

  1. Referee: [§3.2] §3.2: The fast BIC approximation for GMM splits on 1-D projections is load-bearing for the central claim of noise robustness, yet the section provides no analysis or simulation demonstrating that the selected directions correlate with the latent manifold tangent (rather than variance-maximizing noise dimensions) when p ≫ n.

    Authors: We agree that the manuscript would benefit from explicit validation of this point. In the revision we will add a new simulation study in the high-dimensional regime (p ≫ n) that measures the alignment (via cosine similarity or correlation) between the BIC-selected projection directions and the true latent tangent vectors, contrasting them against directions that maximize variance in the noise subspace. This will directly test whether the fast BIC criterion preferentially recovers manifold structure under noise. revision: yes

  2. Referee: [Experiments] Experiments section: The abstract and results claim comparative superiority on simulated and connectome data, but the manuscript supplies neither algorithmic pseudocode, statistical significance tests, error-bar details, nor ablation studies on the sparse-projection and BIC components; this prevents assessment of whether the reported gains are reproducible or attributable to the proposed mechanism.

    Authors: The referee correctly identifies missing elements required for reproducibility. We will add: (i) full pseudocode for URerF in an appendix, (ii) error bars and statistical significance tests (paired Wilcoxon signed-rank tests across repeated runs) for all reported metrics, and (iii) ablation experiments that isolate the contribution of the sparse linear projections and the BIC splitting criterion by comparing against variants that omit each component. revision: yes

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: empirical algorithm validated on external benchmarks

full rationale

The paper introduces URerF as an algorithmic procedure (sparse projections + fast BIC GMM splits) and evaluates it via geodesic precision-recall on simulated manifolds and a connectome dataset. No equations or claims reduce a reported performance quantity to a fitted parameter defined from the same data; the BIC derivation in section 3.2 is an approximation for split selection, not a self-referential prediction. No load-bearing self-citations or uniqueness theorems are invoked. The method is self-contained against external benchmarks (Isomap, UMAP, FLANN) and therefore receives the default non-circularity finding.

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

Abstract supplies insufficient technical detail to enumerate concrete free parameters, axioms, or invented entities.

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Cite this review

Pith. "Pith review of Geodesic Learning via Unsupervised Decision Forests." pith.science (2026). https://pith.science/paper/ZNDA3DKP

@misc{pith2026190702844,
  author       = {Pith},
  title        = {Pith review of: Geodesic Learning via Unsupervised Decision Forests},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZNDA3DKP}},
  note         = {Machine review of arXiv:1907.02844}
}
read the original abstract

Geodesic distance is the shortest path between two points in a Riemannian manifold. Manifold learning algorithms, such as Isomap, seek to learn a manifold that preserves geodesic distances. However, such methods operate on the ambient dimensionality, and are therefore fragile to noise dimensions. We developed an unsupervised random forest method (URerF) to approximately learn geodesic distances in linear and nonlinear manifolds with noise. URerF operates on low-dimensional sparse linear combinations of features, rather than the full observed dimensionality. To choose the optimal split in a computationally efficient fashion, we developed a fast Bayesian Information Criterion statistic for Gaussian mixture models. We introduce geodesic precision-recall curves which quantify performance relative to the true latent manifold. Empirical results on simulated and real data demonstrate that URerF is robust to high-dimensional noise, where as other methods, such as Isomap, UMAP, and FLANN, quickly deteriorate in such settings. In particular, URerF is able to estimate geodesic distances on a real connectome dataset better than other approaches.

Figures

Figures reproduced from arXiv: 1907.02844 by the authors.

Figure 1
Figure 1. Synthetic datasets for all experiments. In each case, there are 1000 points in 3 signal dimensions, as shown. [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. Geodesic recall curves for the three different splitting criteria using both axis-aligned splits (URF; solid lines) and [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Top Geodesic precision versus k for different values of minparent (the smallest splittable node size). Mtry is set to be equal to the square root of the number of features. Bottom Geodesic precision versus k for different values of mtry (the number of features to test at each node). Minparent was set to be equal to 100. Geodesic precision is robust to large variations in these parameters 8 [PITH_FULL_IMAGE:figures/… view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: Top Geodesic precision at k=50 with varying noise dimension from 2 to 10,000, with N = 1000 samples. While all previous state of the art algorithms degrade to chance levels in all settings as the number of noise dimensions increases, URerF never degrades to chance perf…
Figure 5
Figure 5. Figure 5: Left The right Drosophila connectome after adjacency spectral embedding into six-dimensional space, just showing two of the dimensions. Right Geodesic precision versus geodesic recall for various algorithms using cell type as the true label. URerF achieves a higher rec…
Figure 6
Figure 6. Figure 6: Geodesic precision at k=50 with varying noise dimension [PITH_FULL_IMAGE:figures/full_fig_p018_6.png]

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