Consistency of modularity clustering on random geometric graphs

Davis, Erik; Sethuraman, Sunder

doi:10.1214/17-aap1313

Cited by 10 publications

(3 citation statements)

References 78 publications

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“…The consistency of spectral clustering on unsigned graphs for the SBM has been studied in (Lei & Rinaldo, 2015;Sarkar & Bickel, 2015;Rohe et al, 2011) and more recently consistency of several variants of spectral clustering has been shown (Qin & Rohe, 2013;Joseph & Yu, 2016;Chaudhuri et al, 2012;Le et al;Fasino & Tudisco, 2018;Davis & Sethuraman, 2018). Moreover, while the case of multilayer graphs under the SBM has been previously analyzed (Han et al, 2015;Heimlicher et al, 2012;Jog & Loh, 2015;Paul & Chen, 2017;Xu et al, 2014;Yun & Proutiere, 2016), there are no consistency results for matrix power means for multilayer graphs as studied in (Mercado et al, 2018).…”

Section: Consistency Of the Signed Power Mean Laplacian For The Stoch...mentioning

confidence: 99%

Spectral Clustering of Signed Graphs via Matrix Power Means

Mercado,

Tudisco,

Hein

2019

Preprint

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Signed graphs encode positive (attractive) and negative (repulsive) relations between nodes. We extend spectral clustering to signed graphs via the one-parameter family of Signed Power Mean Laplacians, defined as the matrix power mean of normalized standard and signless Laplacians of positive and negative edges. We provide a thorough analysis of the proposed approach in the setting of a general Stochastic Block Model that includes models such as the Labeled Stochastic Block Model and the Censored Block Model. We show that in expectation the signed power mean Laplacian captures the ground truth clusters under reasonable settings where state-of-theart approaches fail. Moreover, we prove that the eigenvalues and eigenvector of the signed power mean Laplacian concentrate around their expectation under reasonable conditions in the general Stochastic Block Model. Extensive experiments on random graphs and real world datasets confirm the theoretically predicted behaviour of the signed power mean Laplacian and show that it compares favourably with state-of-the-art methods.

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Section: Consistency Of the Signed Power Mean Laplacian For The Stoch...mentioning

confidence: 99%

Spectral Clustering of Signed Graphs via Matrix Power Means

Mercado,

Tudisco,

Hein

2019

Preprint

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“…This has motivated theoretical works like [ 67 ] which study the convergence of graph total variation to a continuum weighted total variation (the same paper proposed a topology to study the convergence that didn’t require regularity—in particular pointwise evaluation—of the continuum function). Total variation functionals are also widely used for clustering and segmentation such as in graph cut methods, for example ratio or Cheeger cuts [ 68 , 69 ], graph modularity clustering [ 70 , 71 ], and Ginzburg–Landau segmentation [ 72 – 74 ].…”

Section: Introductionmentioning

confidence: 99%

Rates of convergence for regression with the graph poly-Laplacian

Trillos,

Murray,

Thorpe

2023

Sampl. Theory Signal Process. Data Anal.

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In the (special) smoothing spline problem one considers a variational problem with a quadratic data fidelity penalty and Laplacian regularization. Higher order regularity can be obtained via replacing the Laplacian regulariser with a poly-Laplacian regulariser. The methodology is readily adapted to graphs and here we consider graph poly-Laplacian regularization in a fully supervised, non-parametric, noise corrupted, regression problem. In particular, given a dataset $$\{x_i\}_{i=1}^n$$ { x i } i = 1 n and a set of noisy labels $$\{y_i\}_{i=1}^n\subset \mathbb {R}$$ { y i } i = 1 n ⊂ R we let $$u_n{:}\{x_i\}_{i=1}^n\rightarrow \mathbb {R}$$ u n : { x i } i = 1 n → R be the minimizer of an energy which consists of a data fidelity term and an appropriately scaled graph poly-Laplacian term. When $$y_i = g(x_i)+\xi _i$$ y i = g ( x i ) + ξ i , for iid noise $$\xi _i$$ ξ i , and using the geometric random graph, we identify (with high probability) the rate of convergence of $$u_n$$ u n to g in the large data limit $$n\rightarrow \infty $$ n → ∞ . Furthermore, our rate is close to the known rate of convergence in the usual smoothing spline model.

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“…Consistency results for total variation clustering are presented in [18,11] where the notion of Γ-convergence in a probabilistic setting is used. Consistency results for modularity clustering in a geometric graph setting are presented in [8].…”

Section: Introductionmentioning

confidence: 99%

Gromov–Hausdorff limit of Wasserstein spaces on point clouds

Trillos

2020

Calc. Var.

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We consider a point cloud Xn := {x1, . . . , xn} uniformly distributed on the flat torus T d := R d /Z d , and construct a geometric graph on the cloud by connecting points that are within distance ε of each other. We let P(Xn) be the space of probability measures on Xn and endow it with a discrete Wasserstein distance Wn as introduced independently in [7] and [15] for general finite Markov chains. We show that as long as ε = εn decays towards zero slower than an explicit rate depending on the level of uniformity of Xn, then the space (P(Xn), Wn) converges in the Gromov-Hausdorff sense towards the space of probability measures on T d endowed with the Wasserstein distance.

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Consistency of modularity clustering on random geometric graphs

Cited by 10 publications

References 78 publications

Spectral Clustering of Signed Graphs via Matrix Power Means

Spectral Clustering of Signed Graphs via Matrix Power Means

Rates of convergence for regression with the graph poly-Laplacian

Gromov–Hausdorff limit of Wasserstein spaces on point clouds

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