Topology Learning of Linear Dynamical Systems With Latent Nodes Using Matrix Decomposition
Topology Learning of Linear Dynamical Systems With Latent Nodes Using Matrix Decomposition
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DOI：
10.1109/tac.2021.3124979
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发表时间：
2019-12
影响因子：
6.8
通讯作者：
M. S. Veedu;Harish Doddi;M. Salapaka
中科院分区：
文献类型：
--
作者：
M. S. Veedu;Harish Doddi;M. Salapaka
在这篇文章中，我们提出了一种新的方法来重建网络的线性动力系统的潜在节点的拓扑结构。网络允许有向环和双向边。主要方法依赖于从观测节点获得的功率谱密度矩阵（IPSDM）的逆作为稀疏和低秩矩阵之和的唯一分解。我们提供的条件和方法分解成稀疏和低秩的IPSDM组件。稀疏组件产生与观察到的节点相关联的道德图（MG），并且低秩组件检索隐藏节点的父母，孩子和配偶（马尔可夫毯）。给出了一个给定的反对称矩阵唯一分解为一个稀疏反对称矩阵与一个低秩反对称矩阵之和的充要条件。对于一个大类的系统，观察到的节点，斜对称矩阵的IPSDM的虚部，到稀疏和低秩分量的唯一分解是足够的，以确定观察到的节点的MG以及潜在节点的马尔可夫毯。对于一个大类的系统，所有虚假的链路中所形成的观察节点的MG可以被识别。假设可识别性所需的隐藏节点上的条件，隐藏节点和观察到的节点之间的链接可以被重建，从而从IPSDM检索网络的确切拓扑。此外，对于有限的数据，我们提供了边界上的条目之间的距离的真实和估计的IPSDM。
In this article, we present a novel approach to reconstruct the topology of networked linear dynamical systems with latent nodes. The network is allowed to have directed loops and bi-directed edges. The main approach relies on the unique decomposition of the inverse of power spectral density matrix (IPSDM) obtained from observed nodes as a sum of sparse and low-rank matrices. We provide conditions and methods for decomposing the IPSDM into sparse and low-rank components. The sparse component yields moral graph (MG) associated with the observed nodes, and the low-rank component retrieves parents, children and spouses (the Markov Blanket) of the hidden nodes. The article provides necessary and sufficient conditions for the unique decomposition of a given skew symmetric matrix into sum of a sparse skew symmetric and a low-rank skew symmetric matrices. For a large class of systems, the unique decomposition of imaginary part of the IPSDM of observed nodes, a skew symmetric matrix, into the sparse and the low-rank components is sufficient to identify the MG of the observed nodes as well as the Markov Blanket of latent nodes. For a large class of systems, all spurious links in the MG formed by the observed nodes can be identified. Assuming conditions on hidden nodes required for identifiability, links between hidden and observed nodes can be reconstructed, thus retrieving the exact topology of the network from the IPSDM. Moreover, for finite data, we provide bounds on entry-wise distance between the true and the estimated IPSDMs.