On Adjacency and e-Adjacency in General Hypergraphs: Towards a New e-Adjacency Tensor

Ouvrard, Xavier; Goff, Jean‐Marie Le; Marchand-Maillet, Stéphane

doi:10.1016/j.endm.2018.11.012

Cited by 13 publications

(7 citation statements)

References 5 publications

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“…By substituting the HGFT and IHGFT into (24), any spectral filtering can be implemented in the vertex domain as…”

Section: Hypergraph Filtersmentioning

confidence: 99%

“…Additionally, some other Laplacians for uniform hypergraphs were presented in [22,23], combining hyperedge-wise and pairwise cutting cost functions, respectively. Ouvrard et al [24] proposed e-adjacency tensors for general hypergraphs by first dividing a general hypergraph into multiple layers and then merging all layers by adding additional vertices.…”

Section: Introductionmentioning

confidence: 99%

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Analysis of Hypergraph Signals via High-Order Total Variation

Feng

et al. 2022

Symmetry

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Beyond pairwise relationships, interactions among groups of agents do exist in many real-world applications, but they are difficult to capture by conventional graph models. Generalized from graphs, hypergraphs have been introduced to describe such high-order group interactions. Inspired by graph signal processing (GSP) theory, an existing hypergraph signal processing (HGSP) method presented a spectral analysis framework relying on the orthogonal CP decomposition of adjacency tensors. However, such decomposition may not exist even for supersymmetric tensors. In this paper, we propose a high-order total variation (HOTV) form of a hypergraph signal (HGS) as its smoothness measure, which is a hyperedge-wise measure aggregating all signal values in each hyperedge instead of a pairwise one in most existing work. Further, we propose an HGS analysis framework based on the Tucker decomposition of the hypergraph Laplacian induced by the aforementioned HOTV. We construct an orthonormal basis from the HOTV, by which a new spectral transformation of the HGS is introduced. Then, we design hypergraph filters in both vertex and spectral domains correspondingly. Finally, we illustrate the advantages of the proposed framework by applications in label learning.

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“…By substituting the HGFT and IHGFT into (24), any spectral filtering can be implemented in the vertex domain as…”

Section: Hypergraph Filtersmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Analysis of Hypergraph Signals via High-Order Total Variation

Feng

et al. 2022

Symmetry

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“…Similar to the degree-normalized adjacency matrix in Eq. ( 4), we adopt the well-known normalizing adjacency tensor extension [32],…”

Section: B Tensorized Hypergraph Neural Networkmentioning

confidence: 99%

“…Hence, we aim to extend the proposed model for non-uniform hypergraphs. Motivated by [12] and [32], we propose two methods to extend the uniform hypergraph models to handle nonuniform hypergraphs. Global Node.…”

Section: F Non-uniform Generalizationmentioning

confidence: 99%

LTNN: A Layerwise Tensorized Compression of Multilayer Neural Network

Huang

2019

IEEE Trans. Neural Netw. Learning Syst.

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Hypergraph neural networks (HGNN) have recently become attractive and received significant attention due to their excellent performance in various domains. However, most existing HGNNs rely on first-order approximations of hypergraph connectivity patterns, which ignores important high-order information.To address this issue, we propose a novel adjacency-tensor-based Tensorized Hypergraph Neural Network (THNN). THNN is a faithful hypergraph modeling framework through high-order outer product feature message passing and is a natural tensor extension of the adjacency-matrix-based graph neural networks. The proposed THNN is equivalent to an high-order polynomial regression scheme, which enable THNN with the ability to efficiently extract high-order information from uniform hypergraphs. Moreover, in consideration of the exponential complexity of directly processing high-order outer product features, we propose using a partially symmetric CP decomposition approach to reduce model complexity to a linear degree. Additionally, we propose two simple yet effective extensions of our method for non-uniform hypergraphs commonly found in real-world applications. Results from experiments on two widely used hypergraph datasets for 3-D visual object classification show the promising performance of the proposed THNN.

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“…To represent the topology of hypergraphs, various tensor forms have been proposed in [9][10][11] which represent a hypergraph by a fixed-order tensor. Specifically, all hyperedges in an unweighted hypergraph can be represented by nonzero elements in a fixed-order tensor whose indices denote vertices in each hyperedge.…”

Section: Introductionmentioning

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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On Adjacency and e-Adjacency in General Hypergraphs: Towards a New e-Adjacency Tensor

Cited by 13 publications

References 5 publications

Analysis of Hypergraph Signals via High-Order Total Variation

Analysis of Hypergraph Signals via High-Order Total Variation

LTNN: A Layerwise Tensorized Compression of Multilayer Neural Network

Regularized Recovery by Multi-Order Partial Hypergraph Total Variation

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