Local-Set-Based Graph Signal Sampling and Reconstruction

Gu, Yuantao; Wang, Xiaohan

doi:10.1007/978-3-030-03574-7_7

Cited by 1 publication

(2 citation statements)

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“…of the joint Gaussian distribution of [xp(0), xp(1)] T , defined as and the corresponding P -variate normal distribution of two variables as a product of P distributions as in (30), we can find parameters Q11, Q12, Q21, Q22 which produce the best fitted distribution using the partial derivatives of the log-likelihood function.…”

Section: 4mentioning

confidence: 99%

“…However, in many modern applications, graph topology is not known a priori [1,2,3,4,5,6,7,8,9,10,11,12,13,14], and the focus of this part is therefore on simultaneous estimation of data on a graph and the underlying graph topology. Without loss of generality, it is convenient to assume that the vertices are given, while the edges and their associated weights are part of the solution to the problem considered and need to be estimated from the vertex geometry and/or the observed data [15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31].…”

Section: Introductionmentioning

confidence: 99%

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Graph Signal Processing -- Part III: Machine Learning on Graphs, from Graph Topology to Applications

Stanković¹,

Mandic²,

Daković³

et al. 2020

Preprint

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Many modern data analytics applications on graphs operate on domains where graph topology is not known a priori, and hence its determination becomes part of the problem definition, rather than serving as prior knowledge which aids the problem solution. Part III of this monograph starts by addressing ways to learn graph topology, from the case where the physics of the problem already suggest a possible topology, through to most general cases where the graph topology is learned from the data. A particular emphasis is on graph topology definition based on the correlation and precision matrices of the observed data, combined with additional prior knowledge and structural conditions, such as the smoothness or sparsity of graph connections. For learning sparse graphs (with small number of edges), the least absolute shrinkage and selection operator, known as LASSO is employed, along with its graph specific variant, graphical LASSO. For completeness, both variants of LASSO are derived in an intuitive way, and explained. An in-depth elaboration of the graph topology learning paradigm is provided through several examples on physically well defined graphs, such as electric circuits, linear heat transfer, social and computer networks, and spring-mass systems. As many graph neural networks (GNN) and convolutional graph networks (GCN) are emerging, we have also reviewed the main trends in GNNs and GCNs, from the perspective of graph signal filtering. We have in particular studied the diffusion process over graphs and have shown that the trend of various improvements on GCNs can also be understood from the graph diffusion perspective. Given that the existing GCNs have been introduced largely in a heuristic manner, the definition of different diffusion processes can also serve as a basis for a new design of GCNs. Tensor representation of lattice-structured graphs is next considered, and it is shown that tensors (multidimensional data arrays) are a special class of graph signals, whereby the graph vertices reside on a high-dimensional regular lattice structure. This part of monograph concludes with two emerging applications in financial data processing and underground transportation networks modeling. By means of portfolio cuts of an asset graph, we show how domain knowledge can be meaningfully incorporated into investment analysis. In the underground traffic example, we demonstrate how graph theory can be used to identify the stations in the London underground network which have the greatest influence on the functionality of the traffic, and proceed, in an innovative way, to assess the impact of a station closure on service levels across the city. Contents 1 Introduction 2 2 Geometrically Defined Graph Topologies 3 3 Graph Topology Based on Signal Similarity 4 4 Learning of Graph Laplacian from Data 8 4.1 Imposing Sparsity on the Connection Metric 10 4.2 Smoothness Constrained Learning of Graph Laplacian .

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Section: 4mentioning

confidence: 99%