Pith. sign in

REVIEW 4 major objections 3 minor 48 references

Robust Multi-Manifold Clustering via Simplex Paths

T0 review · 4 major / 3 minor · reviewed 2026-08-06 · deepseek-v4-flash

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

Pith's one-line read Angle-based simplex path metric provably separates intersecting manifolds

desk verdict A genuinely new angle-based simplex path metric for multi-manifold clustering, with strong experiments and a plausible but incomplete theoretical guarantee. read the letter →

arxiv 2507.10710 v1 pith:3FBBHBNW submitted 2025-07-14 stat.ML cs.LG

Haoyu Chen , Anna Little , Akin Narayan This is my paper
classification stat.MLcs.LG MSC 62H3060D0568R10
keywords multi-manifoldclusteringlargestanglepathdistancesimplexgraphdihedralinfinitydenoisingmanifoldlearningsubspace
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 proposes a geometric method for clustering data drawn from several possibly intersecting low-dimensional manifolds. It builds local d-simplices from nearby points, weights connections between adjacent simplices by the dihedral angle between them, and defines the largest angle path distance (LAPD) as the minimum over simplex paths of the worst angle encountered along the path. The central claim is that with a denoising step that removes mixed simplices near intersections, LAPD is small between simplices on the same manifold and large between different manifolds with high probability, so the components can be recovered by distance-based clustering. The concrete guarantee (Theorem 4.7) says this separation holds once the simplex scale e is at least roughly the larger of τ/Θ and (log n/n)^(1/d), with failure probability O($n^{{-(d+1)}}$).

What carries the argument

The central object is the simplex graph G_S: nodes are valid d-simplices (d-dimensional triangles with d+1 vertices) built from annular nearest neighbors, and two simplices are connected when they share d vertices, with edge weight W_S = π - θ, where θ is the dihedral angle between them (or the two-sided variant min{π - θ, θ}). The metric LAPD between two simplices is the minimum over paths of the maximum edge weight along the path, i.e. the flattest route. The argument runs through a gap between the maximal within-manifold LAPD (wLAPD) and the minimal between-manifold LAPD (bLAPD), with a denoising step that discards simplices whose κ-th nearest neighbor LAPD is large, removing mixed simplices near intersections. The load-bearing estimate is Proposition B.4, a single-link volume bound: the set of points that can form an ε-angle link to a given straddle simplex has volume at most C e^d ε τ^(D-d).

What would settle it

Compute, for a fixed straddle simplex formed by points from two linear manifolds intersecting at angle Θ, the volume of points in a tube of radius τ around the second manifold that form an ε-angle link to it, and compare with the bound C e^d ε τ^(D-d) of Proposition B.4; if for small positive τ the measured volume exceeds the predicted constant times that expression, the noisy-case proof collapses. Alternatively, run the full pipeline on two linear manifolds with n large, e near (log n/n)^(1/d), and moderate noise, and check whether the empirical wLAPD-to-bLAPD gap fails at a rate consistent with O($n^{{-(d+1)}}$).

Watch

Extended reading notes

Core claim

The paper's central discovery is that the flattest-path geometry of a simplex graph, rather than any explicit tangent-plane or curvature estimate, is enough to solve multi-manifold clustering. For two linear manifolds intersecting at angle Θ, pure simplices (all vertices on one manifold) can always be joined by paths whose worst dihedral-angle dissimilarity is at most O(τ/e), while any path connecting different manifolds must pass through straddle simplices that make angle at least Θ/4 with both manifolds; a single-link volume estimate shows such straddling chains are unlikely, giving bLAPD ≥ (2C_5 n e^d)^(-1) and, after denoising, bLAPD ≥ Θ/4. Combining the two bounds yields wLAPD ≪ bLAPD with probability 1 - O($n^{{-(d+1)}}$) whenever e ≳ τ/Θ and e ≳ (log n/n)^(1/d). The paper states this as the first high-probability separation guarantee for an angle-based simplex path metric.

Load-bearing premise

The proof depends on a volume estimate, Proposition B.4, for how many points can form an epsilon-angle link to a straddle simplex near the intersection, and that estimate is proved rigorously only in the noiseless case; for positive noise τ the paper supplies only an informal argument in Appendix D, so the high-probability guarantee is not fully established in the noisy setting, and the main theorem also assumes connectivity of the denoised simplex graph and only two linear manifolds.

Editorial extensions

If this is right

  • With n e^d ≳ log n and e above the noise-to-angle ratio, LAPD-based hierarchical clustering recovers the manifold components with failure probability decaying like a power of n.
  • The denoising step makes the between-manifold distance scale like the intersection angle Θ, so even small noise that would otherwise fill the gap does not destroy separability.
  • The number of manifolds can be estimated from the persistence of connected components across scales, removing a required input of many competing algorithms.
  • The algorithm is quasi-linear in n and linear in ambient dimension D (with bounded intrinsic dimension), in contrast to at least quadratic costs for many baselines.
  • Empirically the method handles curvature, noise, and small intersection angles better than the compared subspace and manifold clustering baselines on the tested synthetic and real datasets.

Reading between the lines

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

  • A natural next step left implicit is a curvature-aware upper bound on e; the paper notes curvature forces smaller simplices but does not quantify it, so a testable extension is to verify the predicted τ/Θ versus (log n/n)^(1/d) scale for circles or spheres of varying radius.
  • The two-sided weight improving real-data performance suggests a hybrid metric that mixes angle dissimilarity with point-density or path length could handle low-density intersections better than either alone; the paper itself suggests combining LAPD with a density-based distance.
  • Because the theory treats only two linear manifolds, the hardest open case is multi-way and curved intersections; if the straddle-simplex mechanism extends there, the same machinery would give a general unsupervised segmentation tool for intersecting filamentary or surface data.
  • A testable empirical prediction is that under nonuniform sampling with density bounded above and below, all constants should change but the separation should persist; one can check whether the gap wLAPD ≪ bLAPD remains for strongly nonuniform densities.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 3 minor

Summary. The paper proposes a multi-manifold clustering method based on a graph of d-simplices whose edge weights are dihedral angles, and defines the largest angle path distance (LAPD) as an infinity shortest-path metric on this simplex graph. The authors derive a within-manifold upper bound (Theorem 4.1), a between-manifold lower bound in the raw and denoised settings (Theorems 4.4 and 4.5), and a combined gap theorem (Theorem 4.7) stating that, with high probability, wLAPD is much smaller than bLAPD. The proofs rely on a single-link volume estimate (Proposition B.4) for the volume of points that can form a low-angle link to a straddle simplex. The paper also gives implementation details, a quasi-linear complexity analysis, and extensive experiments on synthetic and real data, including comparisons with SSC, EKSS, LocPCA, PBC, and DCV.

Significance. If fully established, the LAPD separation theorem would be the first high-probability guarantee for an angle-based simplex path metric in multi-manifold clustering, addressing intersecting manifolds with noise. The geometric construction is natural and the noiseless single-link estimate in Lemma D.1 is a genuine technical contribution. The paper also provides a reproducible-style implementation, scalability analysis, and broad experiments, including on real data. However, the central high-probability claim is not fully proved in the noisy setting because Proposition B.4 is only proved for the noiseless case, and several supporting results are asserted or only proved in low dimensions. The contribution is potentially significant, but the main theorem currently outruns its proof.

major comments (4)
  1. [Appendix D, Proposition B.4] The single-link volume estimate is proved rigorously only for the noiseless case (Lemma D.1). The τ>0 case is handled by an informal paragraph that assumes τ≲ε and invokes continuity and a 'rough Taylor series' argument. This is load-bearing: Theorem 4.4 sets ε=(2C5 n e^d)^{-1}, and under the theorem's hypothesis n e^d ≳ log n this ε tends to 0 while τ may be fixed and only constrained by Assumption 3.2 (τ≤e/q). Thus the regime covered by the informal sketch does not include typical parameters satisfying the theorem hypotheses. Theorems 4.4, 4.5, and 4.7 therefore do not currently provide the stated high-probability bLAPD guarantees in the noisy setting. An independent, rigorous derivation of the noisy single-link volume bound is needed.
  2. [Appendix A, Proposition A.1 and Theorem 4.1] The proof of the wLAPD upper bound is complete only for d=1,2 and q=1. For d>2 the text states that the proof is 'straightforward but cumbersome' and is skipped, while the extension to q<1 is supported only by numerical optimization in Figure 9b. Since Theorem 4.1 is stated for general d and q and is used in Theorem 4.7, either a complete proof for all d and q must be supplied, or the theorem must be stated with the restriction under which it is proved.
  3. [Appendix B, Lemma B.3] The proof of Lemma B.3 asserts the triangle inequality ∠(Δ_i, M1) ≤ W^2_S(Δ_i,Δ_{i+1}) + ∠(Δ_{i+1}, M1) without justification. This inequality is the key step that converts a low-weight path into a long chain of straddle simplices, and it is not immediate because W^2_S is an angle between two simplices while ∠(·, M1) is an angle between a simplex and a manifold. A proof or a precise reference for this angle inequality is required for the bLAPD lower bound.
  4. [Theorem 4.7 and connectivity assumptions] The main gap theorem assumes that G_S and G_{S_dns} are connected (Theorem 4.1 and Theorem 4.7), and the proof of Theorem 4.7 says the denoising procedure will not impact wLAPD as long as the graph stays connected. No sampling condition is given that ensures connectivity with high probability. If the graph is disconnected, LAPD is infinite and the separation claim is vacuous. The connectivity requirement should either be proved under the stated sampling assumptions or explicitly incorporated into the theorem hypotheses.
minor comments (3)
  1. [Theorem 4.4 statement vs. proof] The statement of Theorem 4.4 defines C4 = (2C2/(q sin Θ))^{d+1}, but the proof in Appendix B uses C4 = (10C2/(q sin Θ))^{d+1}. These constants should be reconciled.
  2. [Theorem 4.5, denoising parameter κ] The parameter κ = Θ/(4 C5 n e^d) is used as a 'κ-th nearest neighbor' count, which must be an integer. The theorem should specify κ = ⌊Θ C5 n e^d / 4⌋ or otherwise clarify the rounding, since the proof relies on κ ≥ Θ/(8ε).
  3. [Appendix C, proof of Theorem 4.5] The sentence 'there exists a super straddle simplex that ϵ-links to s=Θ/(8ϵ) neighbors... That is, there exists a path of simplices such that Δ1→ϵ,str ...→ϵ,str Δs' is not immediate. The denoising criterion gives κ neighbors within LAPD η, and the argument should explicitly use a prefix of the realizing path to one of those neighbors to obtain a chain of the required length.

Circularity Check

0 steps flagged · score 1.0 of 10

No circular derivation: the LAPD separation theorem is proved from first principles; self-citations to prior work are used as computational tools, not as evidence for the gap bound.

full rationale

The central claim (Theorem 4.7) combines a deterministic wLAPD upper bound (Theorem 4.1, proved in Appendix A) with a probabilistic bLAPD lower bound (Theorems 4.4 and 4.5, proved in Appendices B and C), both derived from the paper's own assumptions and volume estimates. LAPD is defined in Definition 3.1 directly from the simplex graph weights, and the gap result is not an input to the metric's construction. The only author-overlapping citations are Little et al. (2017) for estimating d and tau and Little et al. (2020) for the multi-scale infinity-path-distance approximation; these are implementation tools and are not used to justify the separation theorem, so they are not load-bearing. The paper does contain genuine completeness gaps that a reviewer should weigh as correctness risk, not circularity: Proposition B.4 is proved rigorously only for tau=0 in Lemma D.1, and Appendix D explicitly gives only an informal 'rough Taylor series' argument for tau>0, with the claimed bound vol_D(A_tau) bounded by C e^d epsilon tau^{D-d} asserted rather than proven (the sketch additionally assumes tau less than about epsilon, which the theorem hypotheses do not guarantee). Appendix A's Proposition A.1 is proved only for d=1,2, with larger d supported by numerical optimization, and Lemma B.3's triangle-inequality step is asserted without proof. These gaps affect the completeness of the high-probability claim in the noisy setting but do not make any prediction equivalent to its inputs by construction. Accordingly, no circular step is identified and the circularity score is 1 due only to minor, non-load-bearing self-citation.

Assumptions & free parameters 8 free parameters · 8 assumptions · 0 invented entities

The central claim rests on sampling and geometric-regularity assumptions. The most fragile is the unproved noisy single-link estimate, together with the restriction of the theory to two linear manifolds.

free parameters (8)
  • e (simplex edge scale) = default sqrt(2)*tau
    Controls simplex size. Theory requires e above tau/Theta and above (log n / n)^(1/d), and e <= 1. Implementation sets e to sqrt(2) times the noise level.
  • q (edge distortion threshold) = 1 / (1.25 + 0.15*(d-2))
    Filters elongated simplices. Empirical default; theory uses q to bound simplex edge length ratios.
  • r0 (volume distortion threshold) = theoretical, not used in implementation
    Appears in Assumption 3.1 to enforce volume lower bounds. The paper says tracking simplex volume is unfeasible and that e, q suffice empirically.
  • B (annulus degree) = 25
    Number of annular nearest neighbors. Controls simplex graph size, O(n B^d), and hence scalability.
  • kappa (denoising neighbor count) = 10 log n default; theory uses Theta/(4 C5 n e^d)
    Defines the kappa-th nearest neighbor LAPD used to detect mixed simplices during denoising.
  • eta (denoising threshold) = elbow point of sorted kappaNN LAPDs; theory uses (2 C5 n e^d)^(-1)
    Removes simplices whose kappaNN LAPD is large. The default is a data-dependent elbow heuristic, while the theory requires a specific value involving unknown constants.
  • k (number of scales) = 100
    Number of thresholds in the multi-scale approximation of LAPD and hierarchical clustering.
  • d and tau estimates = estimated via Little et al. 2017 if not provided
    The algorithm can estimate intrinsic dimension and noise level from data, but theory treats them as known inputs.
assumptions (8)
  • domain assumption Assumption 2.1: manifold intersections are (d-1)-dimensional when nonempty.
    Used throughout to define shared faces and dihedral angles between simplices; the paper argues it is a worst-case assumption.
  • domain assumption Assumption 2.2: samples are i.i.d. from a uniform measure on the noisy tube T_tau(M), or from a density bounded away from 0 and infinity.
    Needed for the probabilistic bounds in Theorems 4.4 and 4.5. The paper notes constants change but the result holds for bounded densities.
  • domain assumption Assumption 3.1: a set of valid simplices exists with size constraints e <= edge length <= e/q and volume distortion bounds.
    The algorithm filters simplices to enforce these constraints, but existence of enough valid simplices is assumed for the theory.
  • domain assumption Assumption 3.2: noise level tau is small relative to simplex size, tau <= e/q.
    Controls how much noise can bend a pure simplex and is used in the wLAPD bound and in Lemma B.1.
  • domain assumption Assumption 4.1: the manifold collection is exactly two linear manifolds intersecting at angle Theta.
    The main theorems are proved only in this two-component linear setting; extensions to curvature and more manifolds are deferred.
  • domain assumption Assumption 4.2: volume regularity of the tube near the intersection, vol_D(T_R) in [C1, C2] R tau^(D-d).
    Used in Lemma B.2 to count how many points fall near the intersection and in the path enumeration argument.
  • domain assumption The denoised or full simplex graph G_S is connected.
    Theorem 4.1 and Theorem 4.7 explicitly assume G_S or G_Sdns is connected so all LAPD distances are finite.
  • ad hoc to paper Proposition B.4 single-link volume estimate holds for tau > 0.
    Theorems 4.4 and 4.5 depend on this estimate, but Appendix D proves it only for tau = 0 and gives an informal argument for the noisy case.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Robust Multi-Manifold Clustering via Simplex Paths." pith.science (2026). https://pith.science/paper/3FBBHBNW

@misc{pith2026250710710,
  author       = {Pith},
  title        = {Pith review of: Robust Multi-Manifold Clustering via Simplex Paths},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3FBBHBNW}},
  note         = {Machine review of arXiv:2507.10710}
}
read the original abstract

This article introduces a novel, geometric approach for multi-manifold clustering (MMC), i.e. for clustering a collection of potentially intersecting, d-dimensional manifolds into the individual manifold components. We first compute a locality graph on d-simplices, using the dihedral angle in between adjacent simplices as the graph weights, and then compute infinity path distances in this simplex graph. This procedure gives a metric on simplices which we refer to as the largest angle path distance (LAPD). We analyze the properties of LAPD under random sampling, and prove that with an appropriate denoising procedure, this metric separates the manifold components with high probability. We validate the proposed methodology with extensive numerical experiments on both synthetic and real-world data sets. These experiments demonstrate that the method is robust to noise, curvature, and small intersection angle, and generally out-performs other MMC algorithms. In addition, we provide a highly scalable implementation of the proposed algorithm, which leverages approximation schemes for infinity path distance to achieve quasi-linear computational complexity.

Figures

Figures reproduced from arXiv: 2507.10710 by the authors.

Figure 1
Figure 1. Construction of simplices and simplex paths connected by adjacent simplices over [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. (a) LAPD (indicated by colors) to the target simplex (pink) before de-noising; [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. Default η (cutoff) from the elbow point criteria; figure on the left is for noiseless data, and figure on the right is for the same data with added noise. data, Subsection 5.3 reports results for nonlinear multimanifold clustering on synthetic data, and Subsection 5.4 reports performance on real data. 5.1 Numerical Implementation The pseudo-code of the LAPD algorithm is given in Algorithm 1. Our implementation2 prov… view at source ↗
Figures from the paper (6 more)
Figure 4
Figure 4. Figure 4: Subspace clustering accuracy. risk of a bad path forming is greatly increased by allowing backflipping. Note if no superscript is specified, LAPD refers to the default LAPD1 . 5.2 Subspace Clustering We first evaluate LAPD on a linear subspace clustering task using syn…
Figure 5
Figure 5. Figure 5: Runtime (in seconds) of various clustering methods. (a) Runtime against the [PITH_FULL_IMAGE:figures/full_fig_p019_5.png]
Figure 6
Figure 6. Figure 6: Synthetic manifold learning datasets. Objects on the first row have intrinsic [PITH_FULL_IMAGE:figures/full_fig_p020_6.png]
Figure 7
Figure 7. Figure 7: (a) Optimal configuration for d = 1 that attains the largest wLAPD with con￾straints of noise τ and simplex size e. θ1 corresponds to the (local) optimal configuration for θ = |θ+−θ−| < π 2 ; θ2 corresponds to the (global) optimal configuration for θ = |θ+−θ−| > π 2 . …
Figure 8
Figure 8. Figure 8: Optimal Configuration for d = 2. problem with d + 2 unknowns. The analytical derivation of the solution is straightforward but cumbersome, and the optimal configuration for any dimension d is always when the apex vertices and the shared face appear on the opposite side…
Figure 9
Figure 9. Figure 9: wLAPD upper bound simulation results. In (a), [PITH_FULL_IMAGE:figures/full_fig_p029_9.png]

Discussion (0). Sign in to comment.

Reference graph

Works this paper leans on

48 extracted references · 33 canonical work pages

  1. [1]

    Beyond linear subspace clustering: A comparative study of nonlinear manifold clustering algorithms

    Maryam Abdolali and Nicolas Gillis. Beyond linear subspace clustering: A comparative study of nonlinear manifold clustering algorithms. Computer Science Review, 42: 0 100435, November 2021. ISSN 1574-0137. doi:10.1016/j.cosrev.2021.100435. URL https://www.sciencedirect.com/science/article/pii/S1574013721000757

    arXiv 2021

  2. [2]

    Spectral Clustering Based on Local PCA

    Ery Arias-Castro, Gilad Lerman, and Teng Zhang. Spectral Clustering Based on Local PCA . Journal of Machine Learning Research, 18 0 (9): 0 1--57, 2017. ISSN 1533-7928. URL http://jmlr.org/papers/v18/14-318.html

    work page 2017

  3. [3]

    Multiple Manifold Clustering Using Curvature Constrained Path

    Amir Babaeian, Alireza Bayestehtashk, and Mojtaba Bandarabadi. Multiple Manifold Clustering Using Curvature Constrained Path . PLOS ONE, 10 0 (9): 0 e0137986, September 2015. ISSN 1932-6203. doi:10.1371/journal.pone.0137986. URL https://journals.plos.org/plosone/article?id=10.1371/journal.pone.0137986. Publisher: Public Library of Science

    work page doi:10.1371/journal.pone.0137986 2015

  4. [4]

    Moderate deviation theorem for the Neyman-Pearson statistic in testing uniformity

    R. Basri and D.W. Jacobs. Lambertian reflectance and linear subspaces. IEEE Transactions on Pattern Analysis and Machine Intelligence, 25 0 (2): 0 218--233, 2003. ISSN 0162-8828. doi:10.1109/TPAMI.2003.1177153

    work page Pith review arXiv 2003

  5. [5]

    Laplacian eigenmaps for dimensionality reduction and data representation

    Mikhail Belkin and Partha Niyogi. Laplacian eigenmaps for dimensionality reduction and data representation. Neural Computation, 15 0 (6): 0 1373--1396, 2003. doi:10.1162/089976603321780317

    doi:10.1162/089976603321780317 2003

  6. [6]

    Machine Learning Guide for Oil and Gas Using Python - A Step-by-Step Breakdown with Data, Algorithms, Codes, and Applications

    Hoss Belyadi, Alireza Haghighat, and Doug Yarranton. Machine Learning Guide for Oil and Gas Using Python - A Step-by-Step Breakdown with Data, Algorithms, Codes, and Applications. Elsevier, 2021. ISBN 9780323902321

    work page 2021

  7. [7]

    Bradley and O.L

    P.S. Bradley and O.L. Mangasarian. k- Plane Clustering . Journal of Global Optimization, 16 0 (1): 0 23--32, January 2000. ISSN 1573-2916. doi:10.1023/A:1008324625522. URL https://doi.org/10.1023/A:1008324625522

    doi:10.1023/a:1008324625522

  8. [8]

    A dendrite method for cluster analysis

    Tadeusz Caliński and Harabasz JA. A dendrite method for cluster analysis. Communications in Statistics - Theory and Methods, 3: 0 1--27, 01 1974. doi:10.1080/03610927408827101

    doi:10.1080/03610927408827101 1974

Show all 48 references
  1. [9]

    Camerini

    Paolo M. Camerini. The min-max spanning tree problem and some extensions. Information Processing Letters, 7 0 (1): 0 10--14, 1978

    1978

  2. [10]

    Largest angle path distance for multi-manifold clustering

    Haoyu Chen, Anna Little, and Akil Narayan. Largest angle path distance for multi-manifold clustering. In 2023 International Conference on Sampling Theory and Applications (SampTA), pages 1--7, 2023. doi:10.1109/SampTA59647.2023.10301401

    2023

  3. [11]

    Costeira and T

    J.P. Costeira and T. Kanade. A Multibody Factorization Method for Independently Moving Objects . International Journal of Computer Vision, 29 0 (3): 0 159--179, 1998. ISSN 0920-5691. doi:10.1023/A:1008000628999

    1998 doi

  4. [12]

    The mnist database of handwritten digit images for machine learning research

    Li Deng. The mnist database of handwritten digit images for machine learning research. IEEE Signal Processing Magazine, 29 0 (6): 0 141--142, 2012

    2012

  5. [13]

    Sparse Manifold Clustering and Embedding

    Ehsan Elhamifar and René Vidal. Sparse Manifold Clustering and Embedding . In Advances in Neural Information Processing Systems , volume 24. Curran Associates, Inc., 2011. URL https://proceedings.neurips.cc/paper/2011/hash/fc490ca45c00b1249bbe3554a4fdf6fb-Abstract.html

    2011

  6. [14]

    Sparse Subspace Clustering : Algorithm , Theory , and Applications

    Ehsan Elhamifar and René Vidal. Sparse Subspace Clustering : Algorithm , Theory , and Applications . IEEE Transactions on Pattern Analysis and Machine Intelligence, 35 0 (11): 0 2765--2781, November 2013. ISSN 1939-3539. doi:10.1109/TPAMI.2013.57. Conference Name: IEEE Transac...

    2013 doi

  7. [15]

    Algorithms for two bottleneck optimization problems

    Harold N Gabow and Robert E Tarjan. Algorithms for two bottleneck optimization problems. Journal of Algorithms, 9 0 (3): 0 411--417, 1988

    1988

  8. [16]

    Recommendations for validating hierarchical clustering in consumer sensory projects

    Attila Gere. Recommendations for validating hierarchical clustering in consumer sensory projects. Current Research in Food Science, 6: 0 100522, 2023. ISSN 2665-9271. doi:https://doi.org/10.1016/j.crfs.2023.100522. URL https://www.sciencedirect.com/science/article/pii/S2665927...

    2023

  9. [17]

    Robust multiple manifolds structure learning

    Dian Gong, Xuemei Zhao, and Gérard Medioni. Robust multiple manifolds structure learning. In Proceedings of the 29th International Coference on International Conference on Machine Learning , ICML '12, pages 25--32, Madison, WI, USA, June 2012. Omnipress. ISBN 978-1-4503-1285-1

    2012

  10. [18]

    Metrics and Models for Handwritten Character Recognition

    Trevor Hastie and Patrice Simard. Metrics and Models for Handwritten Character Recognition . In Carlo Malaguerra, Stephan Morgenthaler, and Elvezio Ronchetti, editors, Conference on Statistical Science Honouring the Bicentennial of Stefano Franscini ’s Birth , Monte Verità , p...

    1997 doi

  11. [19]

    Ho, Ming-Husang Yang, Jongwoo Lim, Kuang-Chih Lee, and D

    J. Ho, Ming-Husang Yang, Jongwoo Lim, Kuang-Chih Lee, and D. Kriegman. Clustering appearances of objects under varying illumination conditions. In 2003 IEEE Computer Society Conference on Computer Vision and Pattern Recognition , 2003. Proceedings . , volume 1, pages I--I, Jun...

    2003 arXiv

  12. [20]

    The maximum capacity route problem

    TC Hu. The maximum capacity route problem. Operations Research, 9 0 (6): 0 898--900, 1961

    1961

  13. [21]

    J. J. Hull . A database for handwritten text recognition research. IEEE Transactions on Pattern Analysis and Machine Intelligence, 16 0 (5): 0 550--554, 1994. doi:10.1109/34.291440

    1994 doi

  14. [22]

    Zhang, Hongdong Li, Mathieu Salzmann, and Ian D

    Pan Ji, T. Zhang, Hongdong Li, Mathieu Salzmann, and Ian D. Reid. Deep subspace clustering networks. In Neural Information Processing Systems, 2017. URL https://api.semanticscholar.org/CorpusID:3470712

    2017

  15. [23]

    Learnable subspace clustering

    Jun Li, Hongfu Liu, Zhiqiang Tao, Handong Zhao, and Yun Fu. Learnable subspace clustering. IEEE Transactions on Neural Networks and Learning Systems, 33 0 (3): 0 1119--1133, 2022 a . doi:10.1109/TNNLS.2020.3040379

    2022

  16. [24]

    Zengyi Li, Yubei Chen, Yann LeCun, and Friedrich T. Sommer. Neural manifold clustering and embedding. ArXiv, abs/2201.10000, 2022 b . URL https://api.semanticscholar.org/CorpusID:246275780

    2022 arXiv

  17. [25]

    Subspace clustering using ensembles of K-subspaces

    John Lipor, David Hong, Yan Shuo Tan, and Laura Balzano. Subspace clustering using ensembles of K-subspaces . Information and Inference: A Journal of the IMA, 10 0 (1): 0 73--107, 11 2020. ISSN 2049-8772. doi:10.1093/imaiai/iaaa031. URL https://doi.org/10.1093/imaiai/iaaa031

    2020 doi

  18. [26]

    Anna Little, Mauro Maggioni, and James M. Murphy. Path-based spectral clustering: guarantees, robustness to outliers, and fast algorithms. The Journal of Machine Learning Research, 21 0 (1): 0 6:184--6:249, 2020

    2020

  19. [27]

    Little, Mauro Maggioni, and Lorenzo Rosasco

    Anna V. Little, Mauro Maggioni, and Lorenzo Rosasco. Multiscale geometric methods for data sets i: Multiscale svd, noise and curvature. Applied and Computational Harmonic Analysis, 43 0 (3): 0 504--567, 2017. ISSN 1063-5203. doi:https://doi.org/10.1016/j.acha.2015.09.009. URL ...

    2017 doi

  20. [28]

    G. Liu, Z. Lin, S. Yan, J. Sun, Y. Yu, and Y. Ma. Robust recovery of subspace structures by low-rank representation. IEEE Trans. on Pattern Analysis and Machine Intelligence, 35 0 (1): 0 171--184, 2012

    2012

  21. [29]

    N2d: (not too) deep clustering via clustering the local manifold of an autoencoded embedding

    Ryan McConville, Raúl Santos-Rodríguez, Robert J Piechocki, and Ian Craddock. N2d: (not too) deep clustering via clustering the local manifold of an autoencoded embedding. In 2020 25th International Conference on Pattern Recognition (ICPR), pages 5145--5152, 2021. doi:10.1109/...

    2020

  22. [30]

    Columbia object image library (coil-20)

    Sameer A Nene, Shree K Nayar, Hiroshi Murase, et al. Columbia object image library (coil-20). 1996

    1996

  23. [31]

    Patel and René Vidal

    Vishal M. Patel and René Vidal. Kernel sparse subspace clustering. In 2014 IEEE International Conference on Image Processing (ICIP), pages 2849--2853, 2014. doi:10.1109/ICIP.2014.7025576

    2014

  24. [32]

    Deep subspace clustering

    Xi Peng, Jiashi Feng, Joey Tianyi Zhou, Yingjie Lei, and Shuicheng Yan. Deep subspace clustering. IEEE Transactions on Neural Networks and Learning Systems, 31 0 (12): 0 5509--5521, 2020. doi:10.1109/TNNLS.2020.2968848

    2020

  25. [33]

    Souvenir and R

    R. Souvenir and R. Pless. Manifold clustering. In Tenth IEEE International Conference on Computer Vision ( ICCV '05) Volume 1 , volume 1, pages 648--653 Vol. 1, 2005. doi:10.1109/ICCV.2005.149

    2005 doi

  26. [34]

    Tenenbaum, Vin de Silva, and John C

    Joshua B. Tenenbaum, Vin de Silva, and John C. Langford. A global geometric framework for nonlinear dimensionality reduction. Science, 290 5500: 0 2319--23, 2000. URL https://api.semanticscholar.org/CorpusID:221338160

    2000

  27. [35]

    Estimating the number of clusters in a data set via the gap statistic

    Robert Tibshirani, Guenther Walther, and Trevor Hastie. Estimating the number of clusters in a data set via the gap statistic. Journal of the Royal Statistical Society Series B: Statistical Methodology, 63 0 (2): 0 411--423, 01 2002. ISSN 1369-7412. doi:10.1111/1467-9868.00293...

    2002

  28. [36]

    Tipping and C.M

    M.E. Tipping and C.M. Bishop. Mixtures of probabilistic principal component analyzers. Neural Computation, 11 0 (2): 0 443--482, 1999. ISSN 0899-7667. doi:10.1162/089976699300016728

    1999 doi

  29. [37]

    Tomasi and T

    C. Tomasi and T. Kanade. Shape and motion from image streams under orthography: a factorization method. International Journal of Computer Vision, 9 0 (2): 0 137--154, 1992. ISSN 0920-5691. doi:10.1007/BF00129684

    1992 doi

  30. [38]

    Large sample spectral analysis of graph-based multi-manifold clustering, 2021

    Nicolas Garcia Trillos, Pengfei He, and Chenghui Li. Large sample spectral analysis of graph-based multi-manifold clustering, 2021. URL http://arxiv.org/abs/2107.13610. arXiv:2107.13610 [cs, math, stat]

    2021 arXiv

  31. [39]

    R. Vidal. Subspace clustering. IEEE Signal Processing Magazine, 28 0 (2): 0 52--68, 2011. ISSN 1053-5888. doi:10.1109/MSP.2010.939739

    2011

  32. [40]

    Shankar Sastry

    René Vidal, Yi Ma, and S. Shankar Sastry. Principal Component Analysis . In René Vidal, Yi Ma, and S.S. Sastry, editors, Generalized Principal Component Analysis , Interdisciplinary Applied Mathematics , pages 25--62. Springer, New York, NY, 2016. ISBN 978-0-387-87811-9. doi:1...

    2016 doi

  33. [41]

    Multi-manifold modeling in non-euclidean spaces

    Xu Wang, Konstantinos Slavakis, and Gilad Lerman. Multi-manifold modeling in non-euclidean spaces. In Guy Lebanon and S. V. N. Vishwanathan, editors, Proceedings of the Eighteenth International Conference on Artificial Intelligence and Statistics, volume 38 of Proceedings of M...

    2015

  34. [42]

    Yang, Arvind Ganesh, S

    John Wright, Allen Y. Yang, Arvind Ganesh, S. Shankar Sastry, and Yi Ma. Robust Face Recognition via Sparse Representation . IEEE Transactions on Pattern Analysis and Machine Intelligence, 31 0 (2): 0 210--227, February 2009. ISSN 1939-3539. doi:10.1109/TPAMI.2008.79. Conferen...

    2009 doi

  35. [43]

    Generalized clustering and multi-manifold learning with geometric structure preservation

    Lirong Wu, Zicheng Liu, Jun Xia, Zelin Zang, Siyuan Li, and Stan Z Li. Generalized clustering and multi-manifold learning with geometric structure preservation. In Proceedings of the IEEE/CVF Winter Conference on Applications of Computer Vision, pages 139--147, 2022 a

    2022

  36. [44]

    Deep clustering and visualization for end-to-end high-dimensional data analysis

    Lirong Wu, Lifan Yuan, Guojiang Zhao, Haitao Lin, and Stan Z Li. Deep clustering and visualization for end-to-end high-dimensional data analysis. IEEE Transactions on Neural Networks and Learning Systems, 2022 b

    2022

  37. [45]

    Multi-manifold clustering: A graph-constrained deep nonparametric method

    Xulun Ye and Jieyu Zhao. Multi-manifold clustering: A graph-constrained deep nonparametric method. 93: 0 215--227. ISSN 0031-3203. doi:10.1016/j.patcog.2019.04.029. URL https://www.sciencedirect.com/science/article/pii/S003132031930175X

    2019 doi

  38. [46]

    A Survey of Sparse Representation : Algorithms and Applications

    Zheng Zhang, Yong Xu, Jian Yang, Xuelong Li, and David Zhang. A Survey of Sparse Representation : Algorithms and Applications . IEEE Access, 3: 0 490--530, 2015. ISSN 2169-3536. doi:10.1109/ACCESS.2015.2430359. Conference Name: IEEE Access

    2015

  39. [47]

    Deep adversarial subspace clustering

    Pan Zhou, Yunqing Hou, and Jiashi Feng. Deep adversarial subspace clustering. In 2018 IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 1596--1604, 2018. doi:10.1109/CVPR.2018.00172

    2018

  40. [48]

    Multi-view clustering via multi-manifold regularized non-negative matrix factorization

    Linlin Zong, Xianchao Zhang, Long Zhao, Hong Yu, and Qianli Zhao. Multi-view clustering via multi-manifold regularized non-negative matrix factorization. 88: 0 74--89. ISSN 0893-6080. doi:10.1016/j.neunet.2017.02.003. URL https://www.sciencedirect.com/science/article/pii/S0893...

    2017 doi

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.