Hypergraph Clustering Using a New Laplacian Tensor with Applications in Image Processing

Chang, Jingya; Chen, Yannan; Qi, Liqun; Yan, Hong

doi:10.1137/19m1291601

Cited by 20 publications

(9 citation statements)

References 24 publications

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“…To conclude, we have provided a hypergraph modeling method, and a fast spectral clustering algorithm that is connected to the hypergraph cut problems proposed by (Zhou, Huang, and Schölkopf 2006;Ghoshdastidar and Dukkipati 2015;Saito, Mandic, and Suzuki 2018). A future direction would be to explore other constructions of multi-way similarity which can connect to other uniform and nonuniform hypergraph cuts not having kernel characteristics, such as Laplacian tensor ways (Chen, Qi, and Zhang 2017;Chang et al 2020), total variation and its submodular extension (Hein et al 2013;Yoshida 2019). Also, it would be interesting to study more on connections between this work and a general splitting functions of inhomogeneous cut (Li and Milenkovic 2017;Chodrow, Veldt, and Benson 2021), e.g., to see which class of splitting functions can be connected to the biclique kernel.…”

Section: Discussionmentioning

confidence: 99%

“…There are three variants of this; star (Zhou, Huang, and Schölkopf 2006), clique (Rodriguez 2002;Saito, Mandic, and Suzuki 2018), and inhomogeneous (Li and Milenkovic 2017;Veldt, Benson, and Kleinberg 2020;Liu et al 2021). Other ways are total variation (Hein et al 2013;Li and Milenkovic 2018) and tensor modeling for uniform hypergraph (Hu and Qi 2012;Chen, Qi, and Zhang 2017;Chang et al 2020;Dukkipati 2014, 2015). Our approach follows star and clique ways as well as tensor and its graph reduction approach of (Ghoshdastidar and Dukkipati 2015).…”

Section: Related Workmentioning

confidence: 99%

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Hypergraph Modeling via Spectral Embedding Connection: Hypergraph Cut, Weighted Kernel $k$-means, and Heat Kernel

Saito

2022

Preprint

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We propose a theoretical framework of multi-way similarity to model real-valued data into hypergraphs for clustering via spectral embedding. For graph cut based spectral clustering, it is common to model real-valued data into graph by modeling pairwise similarities using kernel function. This is because the kernel function has a theoretical connection to the graph cut. For problems where using multi-way similarities are more suitable than pairwise ones, it is natural to model as a hypergraph, which is generalization of a graph. However, although the hypergraph cut is well-studied, there is not yet established a hypergraph cut based framework to model multi-way similarity. In this paper, we formulate multi-way similarities by exploiting the theoretical foundation of kernel function. We show a theoretical connection between our formulation and hypergraph cut in two ways, generalizing both weighted kernel k-means and the heat kernel, by which we justify our formulation. We also provide a fast algorithm for spectral clustering. Our algorithm empirically shows better performance than existing graph and other heuristic modeling methods.

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Section: Discussionmentioning

confidence: 99%

Section: Related Workmentioning

confidence: 99%

Hypergraph Modeling via Spectral Embedding Connection: Hypergraph Cut, Weighted Kernel $k$-means, and Heat Kernel

Saito

2022

Preprint

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“…In the k = 3 case, for example, we would have entry a i,j,k = 1 iff {i, j, k} ∈ E. There are several spectral methods for clustering k-uniform hypergraphs, many of which rely on concepts connected to tensor eigenvalues and eigenvectors [39]. Ke et al [34] use a normalized tensor power iteration to compute eigenvectors, while Chang et al [16] take an explicit optimization approach. Hu and Wang [30] derive a clustering method for uniform hypergraphs based on angular separability, and provide several statistical guarantees.…”

Section: Spectral Clustering Methods For Hypergraphsmentioning

confidence: 99%

Nonbacktracking spectral clustering of nonuniform hypergraphs

Chodrow¹,

Eikmeier²,

Haddock³

2022

Preprint

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Spectral methods offer a tractable, global framework for clustering in graphs via eigenvector computations on graph matrices. Hypergraph data, in which entities interact on edges of arbitrary size, poses challenges for matrix representations and therefore for spectral clustering. We study spectral clustering for nonuniform hypergraphs based on the hypergraph nonbacktracking operator. After reviewing the definition of this operator and its basic properties, we prove a theorem of Ihara-Bass type to enable faster computation of eigenpairs. We then propose an alternating algorithm for inference in a hypergraph stochastic blockmodel via linearized belief-propagation, offering proofs that both formalize and extend several previous results. We perform experiments in real and synthetic data that underscore the benefits of hypergraph methods over graph-based ones when interactions of different sizes carry different information about cluster structure. Through an analysis of our algorithm, we pose several conjectures about the limits of spectral methods and detectability in hypergraph stochastic blockmodels writ large.

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“…We measure signal differences among all vertices in each hyperedge by using high-order interactions directly rather than utilizing pairwise interactions extracted from hypergraphs [7,[14][15][16]. We define the initial form of the total variation as…”

Section: Multi-order Total Variation Over a Hypergraphmentioning

confidence: 99%

Regularized Recovery by Multi-Order Partial Hypergraph Total Variation

Feng

et al. 2021

ICASSP 2021 - 2021 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)

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Capturing complex high-order interactions among data is an important task in many scenarios. A common way to model high-order interactions is to use hypergraphs whose topology can be mathematically represented by tensors. Existing methods use a fixed-order tensor to describe the topology of the whole hypergraph, which ignores the divergence of differentorder interactions. In this work, we take this divergence into consideration, and propose a multi-order hypergraph Laplacian and the corresponding total variation. Taking this total variation as a regularization term, we can utilize the topology information contained by it to smooth the hypergraph signal. This can help distinguish different-order interactions and represent high-order interactions accurately.

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Hypergraph Clustering Using a New Laplacian Tensor with Applications in Image Processing

Cited by 20 publications

References 24 publications

Hypergraph Modeling via Spectral Embedding Connection: Hypergraph Cut, Weighted Kernel $k$-means, and Heat Kernel

Hypergraph Modeling via Spectral Embedding Connection: Hypergraph Cut, Weighted Kernel $k$-means, and Heat Kernel

Nonbacktracking spectral clustering of nonuniform hypergraphs

Regularized Recovery by Multi-Order Partial Hypergraph Total Variation

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