Network-topology inference from (vertex) signal observations is a prominent problem across data-science and engineering disciplines. Most existing schemes assume that observations from all nodes are available, but in many practical environments, only a subset of nodes is accessible. A natural (and sometimes effective) approach is to disregard the role of unobserved nodes, but this ignores latent network effects, deteriorating the quality of the estimated graph. Differently, this paper investigates the problem of inferring the topology of a network from nodal observations while taking into account the presence of hidden (latent) variables. Our schemes assume the number of observed nodes is considerably larger than the number of hidden variables and build on recent graph signal processing models to relate the signals and the underlying graph. Specifically, we go beyond classical correlation and partial correlation approaches and assume that the signals are smooth and/or stationary in the sought graph. The assumptions are codified into different constrained optimization problems, with the presence of hidden variables being explicitly taken into account. Since the resulting problems are ill-conditioned and non-convex, the block matrix structure of the proposed formulations is leveraged and suitable convex-regularized relaxations are presented. Numerical experiments over synthetic and real-world datasets showcase the performance of the developed methods and compare them with existing alternatives.

中文翻译：

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从具有隐藏变量的平滑信号和图平稳信号中学习图


根据（顶点）信号观测进行网络拓扑推断是数据科学和工程学科中的一个突出问题。大多数现有方案假设来自所有节点的观察结果都是可用的，但在许多实际环境中，只有节点的子集是可访问的。一种自然的（有时是有效的）方法是忽略未观察到的节点的作用，但这忽略了潜在的网络效应，从而降低了估计图的质量。不同的是，本文研究了从节点观测推断网络拓扑的问题，同时考虑到隐藏（潜在）变量的存在。我们的方案假设观察到的节点数量远大于隐藏变量的数量，并基于最新的图信号处理模型来将信号和底层图关联起来。具体来说，我们超越了经典相关和部分相关方法，并假设信号在所寻求的图中是平滑和/或静止的。这些假设被编码为不同的约束优化问题，并明确考虑隐藏变量的存在。由于产生的问题是病态且非凸的，因此利用所提出的公式的块矩阵结构并提出合适的凸正则化松弛。对合成数据集和真实数据集的数值实验展示了所开发方法的性能，并将其与现有替代方法进行比较。