Signal Processing on Higher-Order Networks: Livin' on the Edge ... and Beyond
TL;DR: In this article, the authors provide a didactic treatment of the emerging topic of signal processing on higher-order networks, with a special emphasis on the concepts needed for the processing of signals supported on these structures.
About: This article is published in Signal Processing.The article was published on 2021-01-14 and is currently open access. It has received 51 citations till now. The article focuses on the topics: Laplacian matrix & Signal processing.
Citations
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TL;DR: In this paper , a growing simplicial network is proposed to model the higher-order interactions represented by clique structures and the scaling constant for Sombor index with the evolution of the network when the initial seed network is modeled as an Erdős-Rényi random graph.
37 citations
Posted Content•
TL;DR: In this paper, a simplicial convolutional neural network (SCNN) is proposed to learn from data defined on simplices, e.g., nodes, edges, triangles, etc.
Abstract: Graphs can model networked data by representing them as nodes and their pairwise relationships as edges. Recently, signal processing and neural networks have been extended to process and learn from data on graphs, with achievements in tasks like graph signal reconstruction, graph or node classifications, and link prediction. However, these methods are only suitable for data defined on the nodes of a graph. In this paper, we propose a simplicial convolutional neural network (SCNN) architecture to learn from data defined on simplices, e.g., nodes, edges, triangles, etc. We study the SCNN permutation and orientation equivariance, complexity, and spectral analysis. Finally, we test the SCNN performance for imputing citations on a coauthorship complex.
10 citations
23 May 2022
TL;DR: In this paper , a simplicial convolutional neural network (SCNN) is proposed to learn from data defined on simplices, e.g., nodes, edges, triangles, etc.
Abstract: Graphs can model networked data by representing them as nodes and their pairwise relationships as edges. Recently, signal processing and neural networks have been extended to process and learn from data on graphs, with achievements in tasks like graph signal reconstruction, graph or node classifications, and link prediction. However, these methods are only suitable for data defined on the nodes of a graph. In this paper, we propose a simplicial convolutional neural network (SCNN) architecture to learn from data defined on simplices, e.g., nodes, edges, triangles, etc. We study the SCNN permutation and orientation equivariance, complexity, and spectral analysis. Finally, we test the SCNN performance for imputing citations on a coauthorship complex.
10 citations
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TL;DR: In this article, the authors proposed a method to co-cluster the vertices and hyperedges of hypergraphs with edge-dependent vertex weights (EDVWs).
Abstract: We propose a novel method to co-cluster the vertices and hyperedges of hypergraphs with edge-dependent vertex weights (EDVWs). In this hypergraph model, the contribution of every vertex to each of its incident hyperedges is represented through an edge-dependent weight, conferring the model higher expressivity than the classical hypergraph. In our method, we leverage random walks with EDVWs to construct a hypergraph Laplacian and use its spectral properties to embed vertices and hyperedges in a common space. We then cluster these embeddings to obtain our proposed co-clustering method, of particular relevance in applications requiring the simultaneous clustering of data entities and features. Numerical experiments using real-world data demonstrate the effectiveness of our proposed approach in comparison with state-of-the-art alternatives.
9 citations
29 Jan 2022
TL;DR: In this paper , the authors proposed convolutional filtering for data whose structure can be modeled by a simplicial complex (SC), where the incidence matrices are used to transfer the signal in adjacent simplices and build a filter bank to jointly filter signals from different levels.
Abstract: This paper proposes convolutional filtering for data whose structure can be modeled by a simplicial complex (SC). SCs are mathematical tools that not only capture pairwise relationships as graphs but account also for higher-order network structures. These filters are built by following the shift-and-sum principle of the convolution operation and rely on the Hodge-Laplacians to shift the signal within the simplex. But since in SCs we have also inter-simplex coupling, we use the incidence matrices to transfer the signal in adjacent simplices and build a filter bank to jointly filter signals from different levels. We prove some interesting properties for the proposed filter bank, including permutation and orientation equivariance, a computational complexity that is linear in the SC dimension, and a spectral interpretation using the simplicial Fourier transform. We illustrate the proposed approach with numerical experiments.
7 citations
References
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Book•
01 Jan 1989
TL;DR: In this paper, the authors provide a thorough treatment of the fundamental theorems and properties of discrete-time linear systems, filtering, sampling, and discrete time Fourier analysis.
Abstract: For senior/graduate-level courses in Discrete-Time Signal Processing. THE definitive, authoritative text on DSP -- ideal for those with an introductory-level knowledge of signals and systems. Written by prominent, DSP pioneers, it provides thorough treatment of the fundamental theorems and properties of discrete-time linear systems, filtering, sampling, and discrete-time Fourier Analysis. By focusing on the general and universal concepts in discrete-time signal processing, it remains vital and relevant to the new challenges arising in the field --without limiting itself to specific technologies with relatively short life spans.
10,388 citations
TL;DR: This survey provides an overview of higher-order tensor decompositions, their applications, and available software.
Abstract: This survey provides an overview of higher-order tensor decompositions, their applications, and available software. A tensor is a multidimensional or $N$-way array. Decompositions of higher-order tensors (i.e., $N$-way arrays with $N \geq 3$) have applications in psycho-metrics, chemometrics, signal processing, numerical linear algebra, computer vision, numerical analysis, data mining, neuroscience, graph analysis, and elsewhere. Two particular tensor decompositions can be considered to be higher-order extensions of the matrix singular value decomposition: CANDECOMP/PARAFAC (CP) decomposes a tensor as a sum of rank-one tensors, and the Tucker decomposition is a higher-order form of principal component analysis. There are many other tensor decompositions, including INDSCAL, PARAFAC2, CANDELINC, DEDICOM, and PARATUCK2 as well as nonnegative variants of all of the above. The N-way Toolbox, Tensor Toolbox, and Multilinear Engine are examples of software packages for working with tensors.
9,227 citations
TL;DR: In this article, the authors present the most common spectral clustering algorithms, and derive those algorithms from scratch by several different approaches, and discuss the advantages and disadvantages of these algorithms.
Abstract: In recent years, spectral clustering has become one of the most popular modern clustering algorithms. It is simple to implement, can be solved efficiently by standard linear algebra software, and very often outperforms traditional clustering algorithms such as the k-means algorithm. On the first glance spectral clustering appears slightly mysterious, and it is not obvious to see why it works at all and what it really does. The goal of this tutorial is to give some intuition on those questions. We describe different graph Laplacians and their basic properties, present the most common spectral clustering algorithms, and derive those algorithms from scratch by several different approaches. Advantages and disadvantages of the different spectral clustering algorithms are discussed.
9,141 citations
Posted Content•
TL;DR: GraphSAGE is presented, a general, inductive framework that leverages node feature information (e.g., text attributes) to efficiently generate node embeddings for previously unseen data and outperforms strong baselines on three inductive node-classification benchmarks.
Abstract: Low-dimensional embeddings of nodes in large graphs have proved extremely useful in a variety of prediction tasks, from content recommendation to identifying protein functions. However, most existing approaches require that all nodes in the graph are present during training of the embeddings; these previous approaches are inherently transductive and do not naturally generalize to unseen nodes. Here we present GraphSAGE, a general, inductive framework that leverages node feature information (e.g., text attributes) to efficiently generate node embeddings for previously unseen data. Instead of training individual embeddings for each node, we learn a function that generates embeddings by sampling and aggregating features from a node's local neighborhood. Our algorithm outperforms strong baselines on three inductive node-classification benchmarks: we classify the category of unseen nodes in evolving information graphs based on citation and Reddit post data, and we show that our algorithm generalizes to completely unseen graphs using a multi-graph dataset of protein-protein interactions.
7,926 citations
TL;DR: In this article, the authors proposed a geometrically motivated algorithm for representing high-dimensional data, based on the correspondence between the graph Laplacian, the Laplace Beltrami operator on the manifold and the connections to the heat equation.
Abstract: One of the central problems in machine learning and pattern recognition is to develop appropriate representations for complex data. We consider the problem of constructing a representation for data lying on a low-dimensional manifold embedded in a high-dimensional space. Drawing on the correspondence between the graph Laplacian, the Laplace Beltrami operator on the manifold, and the connections to the heat equation, we propose a geometrically motivated algorithm for representing the high-dimensional data. The algorithm provides a computationally efficient approach to nonlinear dimensionality reduction that has locality-preserving properties and a natural connection to clustering. Some potential applications and illustrative examples are discussed.
7,210 citations