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Network-topology inference from (vertex) signal observations is a prominentproblem across data-science and engineering disciplines. Most existing schemesassume that observations from all nodes are available, but in many practicalenvironments, only a subset of nodes is accessible. A natural (and sometimeseffective) approach is to disregard the role of unobserved nodes, but thisignores latent network effects, deteriorating the quality of the estimatedgraph. Differently, this paper investigates the problem of inferring thetopology of a network from nodal observations while taking into account thepresence of hidden (latent) variables. Our schemes assume the number ofobserved nodes is considerably larger than the number of hidden variables andbuild on recent graph signal processing models to relate the signals and theunderlying graph. Specifically, we go beyond classical correlation and partialcorrelation approaches and assume that the signals are smooth and/or stationaryin the sought graph. The assumptions are codified into different constrainedoptimization problems, with the presence of hidden variables being explicitlytaken into account. Since the resulting problems are ill-conditioned andnon-convex, the block matrix structure of the proposed formulations isleveraged and suitable convex-regularized relaxations are presented. Numericalexperiments over synthetic and real-world datasets showcase the performance ofthe developed methods and compare them with existing alternatives.

Learning Graphs from Smooth and Graph-Stationary Signals with Hidden Variables