An Intuitive Derivation of the Coherence Index Relation in Compressive Sensing

Stanković, Ljubiša; Mandic, Danilo P.; Daković, Miloš; Kisil, Ilya

doi:10.48550/arxiv.1903.11136

Cited by 2 publications

(4 citation statements)

References 7 publications

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“…where µ(n) is the mean of the observations at the vertex n. Remark 1: Since the correlation matrix in (11) includes contribution from signals at all vertices, it accumulates correlations obtained through all possible walks from the current vertex, n, to any other vertex, m. This also means that the correlation coefficient between two vertices will produce misleading results if there exists one or more other vertices, q, where the signal is strongly correlated with both of the considered vertices, m and n. This is why the naive use of correlation tends to overestimate the strength of direct vertex connections; this renders it a poor metric for establishing direct links (edges) between vertices.…”

Section: Learning Of Graph Laplacian From Datamentioning

confidence: 99%

“…The focus of Part I has been on defining graph properties through the mathematical formalism of linear algebra, while Part II introduced graph counterparts of several important standard data analytics algorithms, again for a given graph. 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: Learning Of Graph Laplacian From Datamentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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

Stanković¹,

Mandic²,

Daković³

et al. 2020

Preprint

Self Cite

View full text Add to dashboard Cite

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“…where µ is equal to the maximum value of the inner product among any two columns of the measurement matrix, A M N (µ is referred to as the coherence index) [47].…”

Section: Unique Reconstruction Conditionsmentioning

confidence: 99%

“…, n M } and (c) If the maximum possible absolute value of µ(k, k i ) is denoted by µ = max|µ(k, k i )| (coherence index of A M N ) then, in the worst case scenario, the amplitude of the largest component, X(k i ), (assumed with the normalized amplitude 1), will be reduced for the maximum possible influence of other equally strong (unity) components 1 − (K − 1)µ, and should be greater than the maximum possible disturbance at k = k i , which is Kµ. From 1 − (K − 1)µ > Kµ, the unique reconstruction condition follows; see also [34,47].…”

Section: Unique Reconstruction Conditionsmentioning

confidence: 99%

Graph Signal Processing -- Part II: Processing and Analyzing Signals on Graphs

Stankovic,

Mandic,

Dakovic

et al. 2019

Preprint

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Data analytics on graphs deals with information processing of data acquired on irregular but structured graph domains. The focus of Part I of this monograph has been on both the fundamental and higher-order graph properties, graph topologies, and spectral representations of graphs. Part I also establishes rigorous frameworks for vertex clustering and graph segmentation, and illustrates the power of graphs in various data association tasks. Part II embarks on these concepts to address the algorithmic and practical issues centered round data/signal processing on graphs, that is, the focus is on the analysis and estimation of both deterministic and random data on graphs. The fundamental ideas related to graph signals are introduced through a simple and intuitive, yet illustrative and general enough case study of multisensor temperature field estimation. The concept of systems on graph is defined using graph signal shift operators, which generalize the corresponding principles from traditional learning systems. At the core of the spectral domain representation of graph signals and systems is the Graph Discrete Fourier Transform (GDFT), which is defined based on the eigendecomposition of both the adjacency matrix and the graph Laplacian. The spectral domain representations are then used as the basis to introduce graph signal filtering concepts and address their design, including Chebyshev polynomial approximation series. Ideas related to the sampling of graph signals, and in particular the challenging topic of data dimensionality reduction through graph subsampling, are presented and further linked with compressive sensing. The principles of time-varying signals on graphs and basic definitions related to random graph signals are next reviewed. Localized graph signal analysis in the joint vertex-spectral domain is referred to as the vertex-frequency analysis, since it can be considered as an extension of classical time-frequency analysis to the graph domain of a signal. Important topics related to the local graph Fourier transform (LGFT) are covered, together with its various forms including the graph spectral and vertex domain windows and the inversion conditions and relations. A link between the LGFT with spectral varying window and the spectral graph wavelet transform (SGWT) is also established. Realizations of the LGFT and SGWT using polynomial (Chebyshev) approximations of the spectral functions are further considered and supported by examples. Finally, energy versions of the vertex-frequency representations are introduced, along with their relations with classical time-frequency analysis, including a vertex-frequency distribution that can satisfy the marginal properties. The material is supported by numerous examples.

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An Intuitive Derivation of the Coherence Index Relation in Compressive Sensing

Cited by 2 publications

References 7 publications

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

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

Graph Signal Processing -- Part II: Processing and Analyzing Signals on Graphs

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