Browsing by Author "Marques, Antonio G."

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    Identifying the Topology of Undirected Networks From Diffused Non-Stationary Graph Signals
    (IEEE, 2021) Shafipour, Rasoul; Segarra, Santiago; Marques, Antonio G.; Mateos, Gonzalo
    We address the problem of inferring an undirected graph from nodal observations, which are modeled as non-stationary graph signals generated by local diffusion dynamics that depend on the structure of the unknown network. Using the so-called graph-shift operator (GSO), which is a matrix representation of the graph, we first identify the eigenvectors of the shift matrix from observations of the diffused signals, and then estimate the eigenvalues by imposing desirable properties on the graph to be recovered. Different from the stationary setting where the eigenvectors can be obtained directly from the covariance matrix of the measurements, here we need to estimate first the unknown diffusion (graph) filter - a polynomial in the GSO that preserves the sought eigenbasis. To carry out this initial system identification step, we exploit different sources of information on the arbitrarily-correlated input signal driving the diffusion on the graph. We first explore the setting where the observations, the input information, and the unknown graph filter are linearly related. We then address the case where the relation is given by a system of matrix quadratic equations, which arises in pragmatic scenarios where only the second-order statistics of the inputs are available. While such a quadratic filter identification problem boils down to a non-convex fourth-order polynomial minimization, we discuss identifiability conditions, propose algorithms to approximate the solution, and analyze their performance. Numerical tests illustrate the effectiveness of the proposed topology inference algorithms in recovering brain, social, financial, and urban transportation networks using synthetic and real-world signals.
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    The Dual Graph Shift Operator: Identifying the Support of the Frequency Domain
    (Springer Nature, 2021) Leus, Geert; Segarra, Santiago; Ribeiro, Alejandro; Marques, Antonio G.
    Contemporary data is often supported by an irregular structure, which can be conveniently captured by a graph. Accounting for this graph support is crucial to analyze the data, leading to an area known as graph signal processing (GSP). The two most important tools in GSP are the graph shift operator (GSO), which is a sparse matrix accounting for the topology of the graph, and the graph Fourier transform (GFT), which maps graph signals into a frequency domain spanned by a number of graph-related Fourier-like basis vectors. This alternative representation of a graph signal is denominated the graph frequency signal. Several attempts have been undertaken in order to interpret the support of this graph frequency signal, but they all resulted in a one-dimensional interpretation. However, if the support of the original signal is captured by a graph, why would the graph frequency signal have a simple one-dimensional support? Departing from existing work, we propose an irregular support for the graph frequency signal, which we coin dual graph. A dual GSO leads to a better interpretation of the graph frequency signal and its domain, helps to understand how the different graph frequencies are related and clustered, enables the development of better graph filters and filter banks, and facilitates the generalization of classical SP results to the graph domain.