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arxiv: 1907.02844 · v1 · pith:ZNDA3DKPnew · submitted 2019-07-05 · 📊 stat.ML · cs.IR· cs.LG· stat.ME

Geodesic Learning via Unsupervised Decision Forests

Meghana Madhyastha , Percy Li , James Browne , Veronika Strnadova-Neeley , Carey E. Priebe , Randal Burns , Joshua T. Vogelstein This is my paper

Pith reviewed 2026-05-25 02:08 UTC · model grok-4.3

classification 📊 stat.ML cs.IRcs.LGstat.ME
keywords geodesic distancesunsupervised random forestsmanifold learninghigh-dimensional noiseBIC for Gaussian mixturesconnectome dataprecision-recall curves
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The pith

Unsupervised random forests recover geodesic distances on noisy high-dimensional manifolds by splitting on sparse feature combinations.

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

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.

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.

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.

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.

Where Pith is reading between the lines

These are 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.

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.

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.

Figures

Figures reproduced from arXiv: 1907.02844 by Carey E. Priebe, James Browne, Joshua T. Vogelstein, Meghana Madhyastha, Percy Li, Randal Burns, Veronika Strnadova-Neeley.

Figure 1
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. 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. 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_IMAG… view at source ↗
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 performance for any of the settings. Bottom Same as top, but each dimension is linearly rescaled to be between 0 and 1, and x-axis shows a small… view at source ↗
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 recall for essentially all precisions. The values of k for this experiment range from 50 to 250 with increments of 50 will perform poorly on re… view at source ↗
Figure 6
Figure 6. Figure 6: Geodesic precision at k=50 with varying noise dimension [PITH_FULL_IMAGE:figures/full_fig_p018_6.png] view at source ↗
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.

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

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

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.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

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

pith-pipeline@v0.9.0 · 5740 in / 1069 out tokens · 27233 ms · 2026-05-25T02:08:13.605697+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

48 extracted references · 48 canonical work pages · 2 internal anchors

  1. [1]

    Top 10 Algorithms in Data Mining,

    X. Wu, V. Kumar, J. Ross Quinlan, J. Ghosh, Q. Y ang, H. Motoda, G. J. McLachlan, A. Ng, B. Liu, P . S. Yu, Z.-H. Zhou, M. Steinbach, D. J. Hand, and D. Steinberg, “Top 10 Algorithms in Data Mining,”Knowledge and information systems, vol. 14, pp. 1–37, Dec. 2007

    work page 2007

  2. [2]

    Consistent Nonparametric Regression,

    C. J. Stone, “Consistent Nonparametric Regression,” Annals of statistics, vol. 5, pp. 595–620, July 1977

    work page 1977

  3. [3]

    Random forests,

    L. Breiman, “Random forests,” Machine learning, vol. 45, no. 1, pp. 5–32, 2001

    work page 2001

  4. [4]

    A decision-theoretic generalization of on-line learning and an application to boosting,

    Y . Freund and R. E. Schapire, “A decision-theoretic generalization of on-line learning and an application to boosting,” Journal of Computer and System Sciences, 1997

    work page 1997

  5. [5]

    An Empirical Comparison of Supervised Learning Algorithms,

    R. Caruana and A. Niculescu-Mizil, “An Empirical Comparison of Supervised Learning Algorithms,” in Pro- ceedings of the 23rd International Conference on Machine Learning , ICML ’06, (New Y ork, NY , USA), pp. 161–168, ACM, 2006

    work page 2006

  6. [6]

    An empirical evaluation of supervised learning in high dimensions,

    R. Caruana, N. Karampatziakis, and A. Y essenalina, “An empirical evaluation of supervised learning in high dimensions,” inProceedings of the 25th international conference on Machine learning, (New Y ork, New Y ork, USA), pp. 96–103, ACM, July 2008

    work page 2008

  7. [7]

    XGBoost: A Scalable Tree Boosting System,

    T. Chen and C. Guestrin, “XGBoost: A Scalable Tree Boosting System,” in Proceedings of the 22Nd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining , KDD ’16, (New Y ork, NY , USA), pp. 785–794, ACM, 2016

    work page 2016

  8. [8]

    ANN-Benchmarks: A Benchmarking Tool for Approximate Nearest Neighbor Algorithms,

    M. Aumüller, E. Bernhardsson, and A. Faithfull, “ANN-Benchmarks: A Benchmarking Tool for Approximate Nearest Neighbor Algorithms,” inSimilarity Search and Applications, pp. 34–49, Springer International Pub- lishing, 2017

    work page 2017

  9. [9]

    Schölkopf and A

    B. Schölkopf and A. J. Smola, Learning with kernels: support vector machines, regularization, optimization, and beyond. MIT press, 2002

    work page 2002

  10. [10]

    Mohri, A

    M. Mohri, A. Rostamizadeh, and A. Talwalkar, Foundations of Machine Learning. MIT Press, Nov. 2018

    work page 2018

  11. [11]

    W. A. Fuller, Measurement Error Models. Wiley, 99 edition ed., June 1987. 11

    work page 1987

  12. [12]

    D. J. Hand, Measurement: A Very Short Introduction (Very Short Introductions) . Oxford University Press, 1 edition ed., Dec. 2016

    work page 2016

  13. [13]

    J. A. Lee and M. Verleysen, Nonlinear Dimensionality Reduction. Springer Science & Business Media, 2007 edition ed., Dec. 2007

    work page 2007

  14. [14]

    Set operations on polyhedra using binary space partitioning trees,

    W. C. Thibault and B. F . Naylor, “Set operations on polyhedra using binary space partitioning trees,” inACM SIGGRAPH computer graphics, vol. 21, pp. 153–162, ACM, 1987

    work page 1987

  15. [16]

    Breiman, J

    L. Breiman, J. Friedman, C. J. Stone, and R. A. Olshen, Classification and Regression Trees (Wadsworth Statistics/Probability). Chapman and Hall/CRC, 1 edition ed., Jan. 1984

    work page 1984

  16. [17]

    Greedy function approximation: a gradient boosting machine,

    J. H. Friedman, “Greedy function approximation: a gradient boosting machine,” Annals of statistics , pp. 1189–1232, 2001

    work page 2001

  17. [18]

    Some infinity theory for predictor ensembles,

    L. Breiman, “Some infinity theory for predictor ensembles,” tech. rep., Technical Report 579, Statistics Dept. UCB, 2000

    work page 2000

  18. [19]

    The Random Forest Kernel and other kernels for big data from random partitions,

    A. Davies and Z. Ghahramani, “The Random Forest Kernel and other kernels for big data from random partitions,” Feb. 2014

    work page 2014

  19. [20]

    Random Forests and Kernel Methods,

    E. Scornet, “Random Forests and Kernel Methods,” IEEE Trans. Inf. Theory, vol. 62, pp. 1485–1500, Mar. 2016

    work page 2016

  20. [21]

    The Mondrian Kernel,

    M. Balog, B. Lakshminarayanan, Z. Ghahramani, D. M. Roy, and Y . W. Teh, “The Mondrian Kernel,” June 2016

    work page 2016

  21. [22]

    Decision Forests Induce Characteristic Kernels,

    C. Shen and J. T. Vogelstein, “Decision Forests Induce Characteristic Kernels,” Nov. 2018

    work page 2018

  22. [23]

    Kernel principal component analysis,

    B. Schölkopf, A. Smola, and K.-R. Müller, “Kernel principal component analysis,” inInternational Conference on Artificial Neural Networks, pp. 583–588, Springer, 1997

    work page 1997

  23. [24]

    Randomer forests,

    T. M. Tomita, M. Maggioni, and J. T. Vogelstein, “Randomer forests,”arXiv preprint arXiv:1506.03410, 2015

    work page arXiv 2015

  24. [25]

    Devroye, L

    L. Devroye, L. Györfi, and G. Lugosi, A Probabilistic Theory of Pattern Recognition (Stochastic Modelling and Applied Probability). Springer, corrected edition ed., Feb. 1997

    work page 1997

  25. [26]

    A global geometric framework for nonlinear dimensionality reduction,

    J. B. Tenenbaum, V. De Silva, and J. C. Langford, “A global geometric framework for nonlinear dimensionality reduction,” science, vol. 290, no. 5500, pp. 2319–2323, 2000

    work page 2000

  26. [27]

    Laplacian eigenmaps and spectral techniques for embedding and clustering,

    M. Belkin and P . Niyogi, “Laplacian eigenmaps and spectral techniques for embedding and clustering,” in Advances in neural information processing systems, pp. 585–591, 2002

    work page 2002

  27. [28]

    UMAP: Uniform Manifold Approximation and Projection for Dimension Reduction

    L. McInnes and J. Healy, “Umap: Uniform manifold approximation and projection for dimension reduction,” arXiv preprint arXiv:1802.03426, 2018

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  28. [29]

    Global Versus Local Methods in Nonlinear Dimensionality Reduction,

    V. D. Silva and J. B. Tenenbaum, “Global Versus Local Methods in Nonlinear Dimensionality Reduction,” in Advances in Neural Information Processing Systems 15 (S. Becker, S. Thrun, and K. Obermayer, eds.), pp. 721–728, MIT Press, 2003

    work page 2003

  29. [30]

    Visualizing Data using t-SNE,

    L. v. d. Maaten and G. Hinton, “Visualizing Data using t-SNE,”J. Mach. Learn. Res., vol. 9, no. Nov, pp. 2579– 2605, 2008

    work page 2008

  30. [31]

    Visualizing data using t-sne,

    L. v. d. Maaten and G. Hinton, “Visualizing data using t-sne,” Journal of machine learning research , vol. 9, no. Nov, pp. 2579–2605, 2008

    work page 2008

  31. [32]

    Scalable nearest neighbor algorithms for high dimensional data,

    M. Muja and D. G. Lowe, “Scalable nearest neighbor algorithms for high dimensional data,” IEEE Transac- tions on Pattern Analysis & Machine Intelligence, no. 11, pp. 2227–2240, 2014. 12

    work page 2014

  32. [33]

    Random projection trees and low dimensional manifolds.,

    S. Dasgupta and Y . Freund, “Random projection trees and low dimensional manifolds.,” in STOC, vol. 8, pp. 537–546, Citeseer, 2008

    work page 2008

  33. [34]

    Random projection trees for vector quantization,

    S. Dasgupta and Y . Freund, “Random projection trees for vector quantization,” IEEE Trans. Inf. Theory , pp. 3229–3242, May 2008

    work page 2008

  34. [35]

    Unsupervised learning with random forest predictors,

    T. Shi and S. Horvath, “Unsupervised learning with random forest predictors,” J. Comput. Graph. Stat. , vol. 15, pp. 118–138, Mar. 2006

    work page 2006

  35. [36]

    Unsupervised learning with random forest predictors,

    T. Shi and S. Horvath, “Unsupervised learning with random forest predictors,” Journal of Computational and Graphical Statistics, vol. 15, no. 1, pp. 118–138, 2006

    work page 2006

  36. [37]

    ROFLMAO: Robust Oblique Forests with Linear MAtrix Opera- tions,

    T. Tomita, M. Maggioni, and J. Vogelstein, “ROFLMAO: Robust Oblique Forests with Linear MAtrix Opera- tions,” in Proceedings of the 2017 SIAM International Conference on Data Mining , Proceedings, pp. 498– 506, Society for Industrial and Applied Mathematics, June 2017

    work page 2017

  37. [38]

    Classification and regression by randomforest,

    A. Liaw and M. Wiener, “Classification and regression by randomforest,” R News, vol. 2, no. 3, pp. 18–22, 2002

    work page 2002

  38. [39]

    Hierarchical Grouping to Optimize an Objective Function,

    J. H. Ward, “Hierarchical Grouping to Optimize an Objective Function,” Journal of the American Statistical Association, vol. 58, pp. 236–244, Mar. 1963

    work page 1963

  39. [40]

    Algorithm AS 136: A K-Means Clustering Algorithm,

    J. A. Hartigan and M. A. Wong, “Algorithm AS 136: A K-Means Clustering Algorithm,” Applied statistics, vol. 28, no. 1, p. 100, 1979

    work page 1979

  40. [41]

    k-means++: The advantages of careful seeding,

    D. Arthur and S. Vassilvitskii, “k-means++: The advantages of careful seeding,” in Proceedings of the eigh- teenth annual ACM-SIAM symposium on Discrete algorithms, pp. 1027–1035, dl.acm.org, 2007

    work page 2007

  41. [42]

    Model-Based Clustering, Discriminant Analysis, and Density Estimation,

    C. Fraley and A. E. Raftery, “Model-Based Clustering, Discriminant Analysis, and Density Estimation,” Jour- nal of the American Statistical Association, vol. 97, pp. 611–631, June 2002

    work page 2002

  42. [43]

    McLachlan and T

    G. McLachlan and T. Krishnan, The EM Algorithm and Extensions . Wiley-Interscience, 2 edition ed., Mar. 2008

    work page 2008

  43. [44]

    Connectal coding: discovering the structures linking cognitive phenotypes to individual histories,

    J. T. Vogelstein, E. W. Bridgeford, B. D. Pedigo, J. Chung, K. Levin, B. Mensh, and C. E. Priebe, “Connectal coding: discovering the structures linking cognitive phenotypes to individual histories,” Current opinion in neurobiology, vol. 55, pp. 199–212, 2019

    work page 2019

  44. [45]

    The complete connectome of a learning and memory centre in an insect brain,

    K. Eichler, F . Li, A. Litwin-Kumar, Y . Park, I. Andrade, C. M. Schneider-Mizell, T. Saumweber, A. Huser, C. Eschbach, B. Gerber, et al., “The complete connectome of a learning and memory centre in an insect brain,”Nature, vol. 548, no. 7666, p. 175, 2017

    work page 2017

  45. [46]

    Semiparametric spectral modeling of the Drosophila connectome

    C. E. Priebe, Y . Park, M. Tang, A. Athreya, V. Lyzinski, J. T. Vogelstein, Y . Qin, B. Cocanougher, K. Eich- ler, M. Zlatic, et al. , “Semiparametric spectral modeling of the drosophila connectome,” arXiv preprint arXiv:1705.03297, 2017

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  46. [47]

    A consistent adjacency spectral embedding for stochastic blockmodel graphs,

    D. L. Sussman, M. Tang, D. E. Fishkind, and C. E. Priebe, “A consistent adjacency spectral embedding for stochastic blockmodel graphs,”Journal of the American Statistical Association, vol. 107, no. 499, pp. 1119– 1128, 2012

    work page 2012

  47. [48]

    Vertex nomination: The canonical sampling and the extended spectral nomination schemes,

    J. Y oder, L. Chen, H. Pao, E. Bridgeford, K. Levin, D. Fishkind, C. Priebe, and V. Lyzinski, “Vertex nomination: The canonical sampling and the extended spectral nomination schemes,” arXiv preprint arXiv:1802.04960, 2018

    work page arXiv 2018

  48. [49]

    Consistency of random forests and other averaging classifiers,

    G. Biau, L. Devroye, and G. Lugosi, “Consistency of random forests and other averaging classifiers,”Journal of Machine Learning Research, vol. 9, no. Sep, pp. 2015–2033, 2008. 13 A Algorithms Algorithm 1 Build an unsupervised random decision tree. Using sparse linear combinations of features for 1-dimensional projections, find the best splitting point among...

    work page 2015