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REVIEW 4 major objections 4 minor 1 cited by

Multi-Community Spectral Clustering for Geometric Graphs

T0 review · 4 major / 4 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read The paper proves that choosing the k-1 adjacency eigenvectors closest to $\lambda^* = n(\mu_{in}-\mu_{out})/k$ and clustering the embedding with $k$-means recovers all $k$ communities in soft geometric block models on the torus.

desk verdict The k-community SGBM extension is real and mostly well-argued, but the proof of Theorem 2.4 is broken as written; worth refereeing, not accepting. read the letter →

arxiv 2508.00893 v1 pith:STT7JGBD submitted 2025-07-27 cs.SI cs.LGmath.PRmath.SPstat.ML

Luiz Emilio Allem , Konstantin Avrachenkov , Carlos Hoppen , Hariprasad Manjunath , Lucas Siviero Sibemberg This is my paper · ORCID
classification cs.SIcs.LGmath.PRmath.SPstat.ML MSC 60B2005C8062H30
keywords SoftGeometricBlockModelCommunitydetectionSpectralclusteringRandomgraphsmatricesDavis-Kahantheoremk-meansembedding
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

This paper extends the soft geometric block model (SGBM) community-detection problem from two communities to any fixed number $k$, and claims that a spectral algorithm succeeds where classical top-eigenvector methods fail. In the SGBM, $n$ nodes are placed uniformly at random on the $d$-dimensional flat torus, and edges appear with probabilities that depend on the distance between nodes and on whether they share a community. The proposed Algorithm 1 forms the $n \times (k-1)$ matrix of eigenvectors belonging to the $k-1$ adjacency eigenvalues closest to $\lambda^* = n(\mu_{in}-\mu_{out})/k$, embeds the rows, and clusters with $k$-means; Theorem 2.3 says this is weakly consistent, with loss $O(\log n / n)$, and that a one-step majority-vote refinement (Algorithm 2) gives perfect recovery with probability tending to 1. The proof rests on a new eigenspace perturbation bound (Theorem 2.4) obtained from a non-standard Davis-Kahan theorem, together with a limiting-spectrum analysis of the adjacency matrix. A sympathetic reader should care because geometric dependence breaks classical spectral clustering, and this is a general method that identifies which eigenvectors actually carry the community label.

What carries the argument

The central object is the population edge-probability matrix $B_\sigma$, the $n \times n$ matrix with $\mu_{in}$ on within-community pairs and $\mu_{out}$ on between-community pairs. Its eigenvalue $\lambda^* = n(\mu_{in}-\mu_{out})/k$ has multiplicity $k-1$, and the corresponding eigenspace has an explicit basis $U$ whose rows are constant on communities, so that $k$-means on any orthonormal basis of that eigenspace separates communities cleanly. The adjacency matrix $A$ is a random perturbation of $B_\sigma$, and the proof transfers the clean eigenspace of $B_\sigma$ to the sample by a non-standard Davis-Kahan theorem that does not require the relevant eigenvalues of $A$ to be simple; it only requires the $k-1$ target eigenvalues to be separated from the rest. The limiting-spectrum result shows the empirical spectral measure converges to a countably supported measure concentrated near the Fourier coefficients of the connectivity functions, which locates $\lambda^*$ as the only cluster carrying community information.

What would settle it

Simulate the SGBM with $k=3$, $n=1000,2000,4000$, and record the within-community degree of one fixed vertex over many position realizations; if its variance exceeds the binomial value $n\mu_{in}(1-\mu_{in})/k$, the Chernoff bound behind Theorem 2.4 cannot hold as written, and the simulation's Algorithm 1 loss will reveal whether the $O(\log n/n)$ rate survives despite the gap.

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Extended reading notes

Core claim

Under the SGBM on the $d$-dimensional flat torus, with $k$ equal-size communities and $\mu_{in} > \mu_{out} > 0$, the paper establishes that the $k-1$ eigenvalues of the adjacency matrix closest to $\lambda^* = n(\mu_{in}-\mu_{out})/k$ are asymptotically almost surely separated by a linear gap from all remaining eigenvalues (Theorems 2.2 and 2.3(a)). If the columns of the matrix $V$ of corresponding orthonormal eigenvectors are used to embed the vertices in $\mathbb{R}^{k-1}$, $k$-means clustering on the rows of $V$ produces an estimator with classification loss at most $\tau \log n / n$ asymptotically almost surely (Theorem 2.3(b)). Replacing each label by the majority community among its graph neighbors turns this into exact recovery, loss $0$, asymptotically almost surely (Theorem 2.3(c)). The quantitative engine is Theorem 2.4: for $U$ the eigenvectors of the population matrix $B_\sigma$ with entries $\mu_{in}$ within communities and $\mu_{out}$ between communities, $\min_{Q \text{ orthogonal}} \|VQ - U\|_F \le \sqrt{12 k^5 \log n}/(\epsilon \sqrt{n})$ asymptotically almost surely, and this bound is what lets a $(1+\epsilon)$-approximate $k$-means solution have few mistakes.

Load-bearing premise

The load-bearing premise is that, in the unconditional SGBM, the entries of the adjacency matrix in a fixed row can be treated as independent Bernoulli variables with fixed means, even though independence actually holds only after conditioning on the random node positions; the proof's Chernoff bounds and the identification of $(A-B_\sigma)U$ with $(A-\mathbb{E}A)U$ depend on that premise.

Editorial extensions

If this is right

  • For any fixed $k$ and any torus dimension $d$, picking the $k-1$ adjacency eigenvectors closest to $\lambda^*$ and running $k$-means gives a weakly consistent community estimator, so the algorithm can be applied without knowing the top of the spectrum.
  • Adding one local majority-vote refinement upgrades weak consistency to strong consistency: after the refinement the partition is exactly the planted one asymptotically almost surely.
  • For the stochastic block model, the theorem reproduces classical spectral clustering: $\lambda^*$ becomes the repeated eigenvalue of the block matrix, and the selected eigenvectors are the usual ones.
  • For the geometric block model with deterministic edge radii, the technical conditions hold except for a measure-zero set of radii, so the algorithm also applies to hard geometric partitions.
  • The Davis-Kahan variant and the auxiliary matrix lemmas are stated generically, so they can be reused for other random-matrix perturbations where the unperturbed eigenvalue is multiple.

Reading between the lines

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

  • We infer that a likely repair of the proof's row-independence gap is to condition on the random positions $X$ and then average, which would preserve the $O(\log n/n)$ rate only if the variance of the unconditional mixture does not dominate; testing this directly with simulated row sums would show whether the theorem's rate is exactly right.
  • We infer that the central mechanism, targeting the mid-spectrum eigenvalue $\lambda^*$ rather than extremal ones, suggests a general design principle for geometric network models: locate the eigenvalue of the population matrix whose eigenspace is constant on latent groups, and use that, not the largest eigenvalues, to embed.
  • We infer that the argument may extend to unequal community sizes by replacing the scalar $\lambda^*$ with a per-community weighted eigenvalue and letting $U$ carry the block sizes; this is an open question the paper lists, and the eigenspace machinery seems adaptable.
  • We infer that if the proof gap is repaired, similar bounds should hold for other compact homogeneous spaces, such as spheres, since the Fourier analysis and the population matrix carry over.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 4 minor

Summary. The paper proposes a spectral clustering algorithm for the soft geometric block model (SGBM) with a fixed number k of communities on the d-dimensional torus, and proves weak and strong consistency of the resulting community estimator. The algorithm selects the k-1 eigenvalues of the adjacency matrix closest to a model-dependent value λ*, forms an embedding from the corresponding eigenvectors, clusters the embedding with k-means, and then applies a local majority-vote refinement. The main technical contributions are a limiting spectral measure for the SGBM (via the moment method) and a Davis-Kahan-type perturbation bound relating the sample eigenvectors to those of the expected block matrix B_σ. The paper extends the earlier k=2 result of Avrachenkov, Bobu, and Dreveton (2021).

Significance. If the main results are correct, the paper provides the first multi-community spectral clustering guarantees for the SGBM, with explicit rates and a perfect-recovery refinement. The limiting spectral analysis and the linear-algebra lemmas (e.g., Lemma 4.5) may be of independent interest. However, the proof of the key perturbation bound (Theorem 2.4) contains several gaps that, although apparently repairable, mean the central claim is not rigorously established as submitted.

major comments (4)
  1. [Section 4, proof of Theorem 2.4] The proof defines X = HU = AU - (EA)U, but since EA = B_σ - μ_in I_n, the correct identity is AU - (EA)U = HU + μ_in U. The bound on ||X||_F therefore controls a different matrix than the one appearing in the Davis-Kahan application. The missing term contributes O(1/n) after division by εn, which is lower-order than the claimed bound, so the issue is repairable, but as written the argument is inconsistent.
  2. [Section 4, equations (36)-(37)] The row sums Y_iz(a) are sums of entries of A, which are independent only conditional on the random positions X. In the unconditional SGBM distribution, these row sums are mixtures of Poisson-binomial variables, so the Chernoff bound of Lemma 4.3 does not apply directly. A two-step concentration argument (first on edges given X, then on X) is needed to justify the stated high-probability bounds.
  3. [Section 4, after equation (39)] The implication that |x_j(i)| > δ only if some single term exceeds its threshold does not follow from the preceding inequality, because the estimate |û_j(p)| < √k together with summation over all k blocks makes the threshold for the sum δ√n/√k, which is smaller than the sum of the individual thresholds Σ δ_p √n. This invalidates the union bound over p as written; the argument can be repaired by using the fact that each column of U has support on only two blocks.
  4. [Theorem 2.3(b) and Section 5] The statement claims that for every τ ∈ (0, ε), ℓ(σ,σ̂) ≤ τ log n / n, but the proof in Theorem 5.1 yields only ℓ ≤ C k^5 log n / n for a constant C that is independent of τ. Since τ can be made arbitrarily small, the stated rate cannot be concluded from the proof; the claim should be weakened to ℓ = O(log n / n) or the proof must be strengthened.
minor comments (4)
  1. [Theorem 2.1] Condition (i) should be stated more precisely as pointwise convergence of the Fourier series of F_in and F_out at 0, rather than the informal 'F_in(0) is equal to the Fourier series evaluated at 0'.
  2. [Section 4] The theorem is restated as 'Theorem 2.-2' in the proof of Theorem 2.4; this is a typo that should be corrected.
  3. [Algorithm 2] Algorithm 2 lists μ_in and μ_out as input but does not use them; the input list should be corrected to include only the output of Algorithm 1.
  4. [Lemma 5.1] The notation for the matrix V with a bar is not clearly distinguished from V in the typeset text; consider using a different symbol or clarifying the typesetting.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the proof derives the spectral measure and perturbation bounds from the model; self-citations are technical and not load-bearing.

full rationale

The paper's derivation chain is self-contained against its model assumptions. Theorem 2.1 (limiting spectral measure) is proved directly by the method of moments, expanding trace moments of A in terms of convolutions of Fin and Fout and counting closed walks; no fitted quantity is renamed as a prediction. Theorem 2.2 follows from that limit measure plus non-degeneracy conditions (10)-(11). Theorem 2.3 then rests on Theorem 2.4, a Davis-Kahan perturbation bound, and Lemma 5.1, a standard k-means perturbation argument. The parameters mu_in and mu_out are inputs of the model, not fitted to the recovered labels, and the loss bound is not an assumed output. The paper cites [3] (whose first author overlaps with the present authors) for the SGBM model, Theorem 1.1, and technical lemmas, but these are auxiliary and do not assume the k>=2 consistency result being proved; the adaptation Lemma A.2 re-proves the genericity condition rather than importing the conclusion. There is no step in which the target claim is used as a premise or in which a fitted parameter is called a prediction. The notable weaknesses in the proof of Theorem 2.4 - equating (A-B_sigma)U with AU-(EA)U despite EA=B_sigma-mu_in I, and applying Chernoff to row sums that are independent only conditional on the random positions - are correctness gaps, not circular reductions: the claimed bound would be unsupported as written, but it is not equivalent to the theorem's input by construction.

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

The central results rest on the model definition, the Fourier-series condition (i), the non-degeneracy conditions (ii)-(iii), and the concentration arguments. No numerical parameter is fitted to data; µ_in and µ_out are inputs. The main unproven support is the unconditional independence assumption for row sums, which is used to derive the perturbation bound.

assumptions (4)
  • ad hoc to paper The technical conditions (i)-(iii) of Theorem 2.3 hold, in particular F_in(0)=F̂_in(0) and F_out(0)=F̂_out(0).
    Required for the identity F^{*m}(0)=∑_z F̂(z)^m used in the moment computation in the proof of Theorem 2.1. This condition excludes hard-threshold geometric block models such as the indicator kernels used in Example 1.1, despite the paper's claim that Lemma A.2 covers GBM.
  • ad hoc to paper The edge indicators A_ij in a fixed row are independent in the unconditional distribution of the SGBM.
    The proof of Theorem 2.4 applies the Chernoff bound (Lemma 4.3) to row sums Y_iz(a) with the unconditional expectation E[Y], but edges are independent only conditional on the random positions X. This assumption is false as stated; a two-step concentration argument would be needed. It is load-bearing for the bound on ||(A-EA)U||_F.
  • domain assumption The Davis-Kahan theorem in the form of Theorem 4.1 applies with M=B_σ and M̃=A despite the diagonal shift -µ_in in the expectation of A.
    The eigen-gap between λ* and the rest of the spectrum is nµ_out + O(1), so the theorem applies for large n. The proof's X=HU=AU-(EA)U substitution is algebraically incorrect; the correct relation is (A-B_σ)U = (A-EA)U - µ_in U, which still gives the same order of bound.
  • domain assumption Positions X_i are i.i.d. uniform on the flat torus and community sizes are exactly n/k.
    The model definition and the theorem statement use this to compute expectations and row concentration bounds.

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Pith. "Pith review of Multi-Community Spectral Clustering for Geometric Graphs." pith.science (2026). https://pith.science/paper/STT7JGBD

@misc{pith2026250800893,
  author       = {Pith},
  title        = {Pith review of: Multi-Community Spectral Clustering for Geometric Graphs},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/STT7JGBD}},
  note         = {Machine review of arXiv:2508.00893}
}
abstract

In this paper, we consider the soft geometric block model (SGBM) with a fixed number $k \geq 2$ of homogeneous communities in the dense regime, and we introduce a spectral clustering algorithm for community recovery on graphs generated by this model. Given such a graph, the algorithm produces an embedding into $\mathbb{R}^{k-1}$ using the eigenvectors associated with the $k-1$ eigenvalues of the adjacency matrix of the graph that are closest to a value determined by the parameters of the model. It then applies $k$-means clustering to the embedding. We prove weak consistency and show that a simple local refinement step ensures strong consistency. A key ingredient is an application of a non-standard version of Davis-Kahan theorem to control eigenspace perturbations when eigenvalues are not simple. We also analyze the limiting spectrum of the adjacency matrix, using a combination of combinatorial and matrix techniques.

Figures

Figures reproduced from arXiv: 2508.00893 by the authors.

Figure 1
Figure 1. The points in R 3 that correspond to the elements in the setting of Example 1.1 for the eigenvectors v4, v5 and v6. Elements in the same community have been drawn with the same color. associated with the adjacency matrix of A. Consider the 1000 × 3 matrix V whose columns are unit eigenvectors v4, v5, and v6 associated with the eigenvalues λ4 ≈ 163.37, λ5 ≈ 162.75, and λ6 ≈ 160.65, respectively, which are the three e… view at source ↗
Figure 2
Figure 2. The points in R 3 that correspond to the elements in the setting of Example 1.1 if the columns of U were replaced by v2, v3 and v4. Elements in the same community have been drawn with the same color. 2. Statement of the main result Recall that Td = R d/Z d is the d dimensional flat unit torus. We follow the definition of a Soft Geometric Block Model given in Section 1.1. For a measurable function φ: Td → R, we consi… view at source ↗
Figure 3
Figure 3. Ternary tree starting with d1, corresponding to binary digits 101. start with a root which is labeled by an element of {d1, . . . , dk}, say d1. To define the next layer we consider b1. If b1 = 0, we connect the root to a single child and with the same label. If b1 = 1, we connect it to k − 1 children, each with one of the labels that is different from their parent. Now, suppose we already defined the layer l of k-a… view at source ↗

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Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Spectral graph clustering with inhomogeneous latent geometry

    cs.SI 2026-08 conditional novelty 7.0 of 10

    Communities in graphs with inhomogeneous latent geometry can be recovered by density-based spectral clustering on deeper eigenspaces, with a proven almost-exact recovery guarantee.

Reference graph

Works this paper leans on

35 extracted references · 35 canonical work pages · cited by 1 Pith paper

  1. [1]

    177, 1–86

    Emmanuel Abbe,Community detection and stochastic block models: recent developments, Journal of Machine Learning Research18(2018), no. 177, 1–86

    work page 2018

  2. [2]

    1, 109– 160

    Emmanuel Abbe, Fran¸ cois Baccelli, and Abishek Sankararaman,Community detection on Eu- clidean random graphs, Information and Inference: A Journal of the IMA10(2020), no. 1, 109– 160

    work page 2020

  3. [3]

    Konstantin Avrachenkov, Andrei Bobu, and Maximilien Dreveton,Higher-order spectral clustering for geometric graphs, Journal of Fourier Analysis and Applications27(2021), no. 2, 22

    work page 2021

  4. [4]

    Konstantin Avrachenkov and Maximilien Dreveton,Statistical analysis of networks, Now Publish- ers, 2022

    work page 2022

  5. [5]

    Konstantin Avrachenkov, BR Kumar, and Lasse Leskel¨ a,Community detection on block models with geometric kernels, ArXiv preprint arXiv:2403.02802 (2024)

    work page arXiv 2024

  6. [6]

    20, Springer, 2010

    Zhidong Bai and Jack W Silverstein,Spectral analysis of large dimensional random matrices, vol. 20, Springer, 2010

    work page 2010

  7. [7]

    4, 515–532

    Charles Bordenave,Eigenvalues of Euclidean random matrices, Random Structures & Algorithms 33(2008), no. 4, 515–532

    work page 2008

  8. [8]

    3-5, 75–174

    Santo Fortunato,Community detection in graphs, Physics reports486(2010), no. 3-5, 75–174

    work page 2010

Show all 35 references
  1. [9]

    32, 2018

    Sainyam Galhotra, Arya Mazumdar, Soumyabrata Pal, and Barna Saha,The geometric block model, Proceedings of the AAAI Conference on Artificial Intelligence, vol. 32, 2018

    2018

  2. [10]

    338, 1–53

    ,Community recovery in the geometric block model, Journal of Machine Learning Research 24(2023), no. 338, 1–53

    2023

  3. [11]

    Gao, Dominic Dwyer, Ye Zhu, Catherine L

    Caroline X. Gao, Dominic Dwyer, Ye Zhu, Catherine L. Smith, Lan Du, Kate M. Filia, Johanna Bayer, Jana M. Menssink, Teresa Wang, Christoph Bergmeir, Stephen Wood, and Sue M. Cot- ton,An overview of clustering methods with guidelines for application in mental health research, P...

    2023

  4. [12]

    Guan,Sharp exact recovery threshold for two-community Euclidean random graphs, ISIT, 2025

    Julia Gaudio and Charlie K. Guan,Sharp exact recovery threshold for two-community Euclidean random graphs, ISIT, 2025

    2025

  5. [13]

    2158–2184

    Julia Gaudio, Xiaochun Niu, and Ermin Wei,Exact community recovery in the geometric SBM, SODA, 2024, pp. 2158–2184

    2024

  6. [14]

    161– 248

    Loukas Grafakos,Fourier analysis on the torus, Classical Fourier Analysis, Springer, 2008, pp. 161– 248

    2008

  7. [15]

    Holland, Kathryn Blackmond Laskey, and Samuel Leinhardt,Stochastic blockmodels: First steps, Social Networks5(1983), no

    Paul W. Holland, Kathryn Blackmond Laskey, and Samuel Leinhardt,Stochastic blockmodels: First steps, Social Networks5(1983), no. 2, 109–137

    1983

  8. [16]

    Roger A Horn and Charles R Johnson,Matrix analysis, Cambridge university press, 2012

    2012

  9. [17]

    Jain,Data clustering: 50 years beyond k-means, Pattern Recognition Letters31(2010), no

    Anil K. Jain,Data clustering: 50 years beyond k-means, Pattern Recognition Letters31(2010), no. 8, 651–666, Award winning papers from the 19th International Conference on Pattern Recog- nition (ICPR)

    2010

  10. [18]

    Amit Kumar, Yogish Sabharwal, and Sandeep Sen,A simple linear time(1 +ϵ)-approximation al- gorithm fork-means clustering in any dimensions, 45th Annual IEEE Symposium on Foundations of Computer Science, IEEE, 2004, pp. 454–462

    2004

  11. [19]

    Lee, Shayan Oveis Gharan, and Luca Trevisan,Multiway spectral partitioning and higher- order cheeger inequalities, J

    James R. Lee, Shayan Oveis Gharan, and Luca Trevisan,Multiway spectral partitioning and higher- order cheeger inequalities, J. ACM61(2014), no. 6

    2014

  12. [20]

    Jing Lei and Alessandro Rinaldo,Consistency of spectral clustering in stochastic block models, The Annals of Statistics (2015), 215–237

    2015

  13. [21]

    eigenspace and singular subspace variations, SIAM Journal on Matrix Analysis and Applications20(1998), no

    Ren-Cang Li,Relative perturbation theory: Ii. eigenspace and singular subspace variations, SIAM Journal on Matrix Analysis and Applications20(1998), no. 2, 471–492

    1998

  14. [22]

    Macqueen,Some methods for classification and analysis of multivariate observations, In 5-th Berkeley Symposium on Mathematical Statistics and Probability (1967), 281–297

    J. Macqueen,Some methods for classification and analysis of multivariate observations, In 5-th Berkeley Symposium on Mathematical Statistics and Probability (1967), 281–297

    1967

  15. [23]

    Meena Mahajan, Prajakta Nimbhorkar, and Kasturi Varadarajan,The planar k-means problem is np-hard, Theoretical computer science442(2012), 13–21

    2012

  16. [24]

    R. E. Mansano, L. E. Allem, R. R. Del-Vecchio, and C. Hoppen,Balanced portfolio via signed graphs and spectral clustering in the Brazilian stock market, Quality & Quantity (2021), 1–16. 30

    2021

  17. [25]

    Michael Mitzenmacher and Eli Upfal,Probability and computing: Randomization and probabilistic techniques in algorithms and data analysis, Cambridge university press, 2017

    2017

  18. [26]

    M. E. J. Newman,Modularity and community structure in networks, Proceedings of the National Academy of Sciences103(2006), no. 23, 8577–8582

    2006

  19. [27]

    A. Y. Ng, M. I. Jordan, and Y. Weiss,On spectral clustering: Analysis and an algorithm, Pro- ceedings of the 14th International Conference on Neural Information Processing Systems: Natural and Synthetic (2001), 849–856

    2001

  20. [28]

    Qin and L

    G. Qin and L. Gao,Spectral clustering for detecting protein complexes in protein–protein interac- tion (PPI) networks, Mathematical and Computer Modelling52(2010), 2066–2074

    2010

  21. [29]

    2181–2200

    Abishek Sankararaman and Fran¸ cois Baccelli,Community detection on Euclidean random graphs, Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms, SIAM, 2018, pp. 2181–2200

    2018

  22. [30]

    Sibson,Slink: An optimally efficient algorithm for the single-link cluster method, The Computer Journal16(1973), no

    R. Sibson,Slink: An optimally efficient algorithm for the single-link cluster method, The Computer Journal16(1973), no. 1, 30–34

    1973

  23. [31]

    Spielman and Shang-Hua Teng,Spectral partitioning works: Planar graphs and finite element meshes, Linear Algebra and its Applications421(2007), no

    Daniel A. Spielman and Shang-Hua Teng,Spectral partitioning works: Planar graphs and finite element meshes, Linear Algebra and its Applications421(2007), no. 2, 284–305, Special Issue in honor of Miroslav Fiedler

    2007

  24. [32]

    Michel Talagrand,A new look at independence, The Annals of probability (1996), 1–34

    1996

  25. [33]

    2, 555–586

    Ulrike Von Luxburg, Mikhail Belkin, and Olivier Bousquet,Consistency of spectral clustering, Annals of Statistics36(2008), no. 2, 555–586

    2008

  26. [34]

    Ulrike von Luxburg, Robert C. Williamson, and Isabelle Guyon,Clustering: Science or art?, Proceedings of ICML Workshop on Unsupervised and Transfer Learning (Bellevue, Washington, USA) (Isabelle Guyon, Gideon Dror, Vincent Lemaire, Graham Taylor, and Daniel Silver, eds.), Proc...

    2012

  27. [35]

    C. Wei, X. Yao, D. Gong, and H. Liu,Spectral clustering based mutant reduction for mutation testing, Information and Software Technology132(2021). AppendixA.Auxiliary Results Theorem A.1.Letb= (b 1,...,b m)be a binary tuple and letX m k ={(x 1,...,x m) :x i∈ {d1,...,d k}}be th...

    2021

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