The self-consistent field iteration for p-spectral clustering

Upadhyaya, Parikshit; Jarlebring, Elias; Tudisco, Francesco

doi:10.48550/arxiv.2111.09750

Cited by 3 publications

(3 citation statements)

References 30 publications

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“…For example, similar to the original Euclidean p-Laplacian and graph linear Laplacian, the p-Laplacian on graphs has some important relations to Cheeger cut problem and shortest path problem on graphs. Just as the Laplacian matrix which has been successfully used in diverse areas, the graph p-Laplacian has been also widely used in various applications, including spectral clustering [10,32,55,56], data and image processing problems, semi-supervised learning and unsupervised learning [35,55,56]. Much recent work has shown that algorithms based on the graph p-Laplacian perform better than classical algorithms based on the linear Laplacian in solving these practical problems in image science.…”

Section: Introductionmentioning

confidence: 99%

Nodal domain theorems for $p$-Laplacians on signed graphs

Ge,

Liu,

Zhang

2023

J. Spectr. Theory

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We establish various nodal domain theorems for p-Laplacians on signed graphs, which unify most of the existing results on nodal domains of graph p-Laplacians and arbitrary symmetric matrices. Based on our nodal domain estimates, we obtain a higher order Cheeger inequality that relates the variational eigenvalues of p-Laplacians and Atay-Liu's multi-way Cheeger constants on signed graphs. In the particular case of p D 1, this leads to several identities relating variational eigenvalues and multi-way Cheeger constants. Intriguingly, our approach also leads to new results on usual graphs, including a weak version of Sturm's oscillation theorem for graph 1-Laplacians and nonexistence of eigenvalues between the largest and second largest variational eigenvalues of p-Laplacians with p > 1 on connected bipartite graphs.

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Section: Introductionmentioning

confidence: 99%

Nodal domain theorems for $p$-Laplacians on signed graphs

Ge,

Liu,

Zhang

2023

J. Spectr. Theory

View full text Add to dashboard Cite

show abstract

“…For example, similar to the original Euclidean p-Laplacian and graph linear Laplacian, the p-Laplacian on graphs has some important relations to Cheeger cut problem and shortest path problem on graphs. Just as the Laplacian matrix which has been successfully used in diverse areas, the graph p-Laplacian has been also widely used in various applications, including spectral clustering [7,16,30,31], data and image processing problems, semi-supervised learning and unsupervised learning [27,30,31]. Much recent work has shown that algorithms based on the graph p-Laplacian perform better than classical algorithms based on the linear Laplacian in solving these practical problems in image science.…”

Section: Introductionmentioning

confidence: 99%

Nodal domain theorems for $p$-Laplacians on signed graphs

Ge¹,

Liu²,

Zhang³

2022

Preprint

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We establish various nodal domain theorems for generalized p-Laplacians on signed graphs, which unify most of the results on nodal domains of graph p-Laplacians and arbitrary symmetric matrices. Based on our nodal domain estimates, we also obtain a higher order Cheeger inequality that relates the variational eigenvalues of p-Laplacians and Atay-Liu's multi-way Cheeger constants on signed graphs. Moreover, we show new results for 1-Laplacian on signed graphs, including a sharp upper bound of the number of nonzeros of certain eigenfunction corresponding to the k-th variational eigenvalues, and some identities relating variational eigenvalues and k-way Cheeger constants.

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“…It is particularly interesting that this type of nonlinear Laplacian operator appears in many settings. For example, in the graph case, if f = id and g(x) = |x| p−1 sign(x), then L boils down to the graph p-Laplacian operator [16,23,42,52]. Exponential-and logarithmicbased choices of f and g give rise to nonlinear Laplacians used to model chemical reactions [37,53] as well as to model consensus dynamics and opinion formation in hypergraphs [35].…”

mentioning

confidence: 99%

Core-periphery detection in hypergraphs

Tudisco¹,

Higham²

2022

Preprint

Self Cite

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Core-periphery detection is a key task in exploratory network analysis where one aims to find a core, a set of nodes well-connected internally and with the periphery, and a periphery, a set of nodes connected only (or mostly) with the core. In this work we propose a model of core-periphery for higher-order networks modeled as hypergraphs and we propose a method for computing a core-score vector that quantifies how close each node is to the core. In particular, we show that this method solves the corresponding non-convex core-periphery optimization problem globally to an arbitrary precision. This method turns out to coincide with the computation of the Perron eigenvector of a nonlinear hypergraph operator, suitably defined in term of the incidence matrix of the hypergraph, generalizing recently proposed centrality models for hypergraphs. We perform several experiments on synthetic and real-world hypergraphs showing that the proposed method outperforms alternative core-periphery detection algorithms, in particular those obtained by transferring established graph methods to the hypergraph setting via clique expansion.

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The self-consistent field iteration for p-spectral clustering

Cited by 3 publications

References 30 publications

Nodal domain theorems for $p$-Laplacians on signed graphs

Nodal domain theorems for $p$-Laplacians on signed graphs

Nodal domain theorems for $p$-Laplacians on signed graphs

Core-periphery detection in hypergraphs

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