This paper considers the problem of learning the unknown structure of a network with the underlying topology given by a polyforest (a collection of directed trees with potentially multiple roots). The main result is an algorithm that consistently learns the network structure using only second-order statistics of the data. The methodology is robust with respect to the presence of unmeasured (latent) nodes: the algorithm detects the exact number and location of the latent nodes, when they satisfy specific degree conditions in the actual network graph. It is shown that the same degree conditions are also necessary for a consistent reconstruction. Thus, the proposed reconstruction algorithm achieves the fundamental limitations in learning the structure of a polyforest network of linear dynamic systems in the presence of latent nodes. This paper overcomes the limitations of previous results that only addressed single-rooted trees, tackling the problem in an efficient way since the computational complexity of the derived algorithm is proven to be polynomial in the number of observed nodes.

中文翻译：

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存在隐藏节点的树状网络拓扑的盲学习

本文考虑了学习网络的未知结构的问题，该网络具有由多森林（具有潜在多个根的有向树的集合）给出的底层拓扑。主要结果是一种算法，该算法仅使用数据的二阶统计量一致地学习网络结构。该方法对于未测量（潜在）节点的存在是稳健的：当潜在节点满足实际网络图中的特定度数条件时，该算法会检测潜在节点的确切数量和位置。结果表明，相同的度条件对于一致的重建也是必要的。因此，所提出的重建算法实现了在存在潜在节点的情况下学习线性动态系统的多森林网络结构的基本限制。