基于部分连通性信息的流静态图信号在线拓扑推断
Online Topology Inference from Streaming Stationary Graph Signals with Partial Connectivity Information
摘要：
我们开发在线图学习算法从流网络数据。我们的目标是跟踪（可能）随时间变化的网络拓扑，并通过在获取数据时实时处理数据来节省内存和计算。该设置需要观测建模为由未知网络上的局部扩散动力学生成的静态图信号。此外，我们可能有先验信息的存在或不存在的一些边缘，如在链接预测问题。平稳性假设意味着观测值的协方差矩阵和所谓的图移位算子（GSO -编码图拓扑的矩阵）在温和的要求下交换。这促使将拓扑推理任务制定为逆问题，由此搜索结构上可接受并且与观测的经验协方差矩阵近似交换的稀疏GSO。对于流数据说，协方差可以递归更新，我们表明，在线近端梯度迭代可以有效地跟踪时变解的逆问题与量化的保证。具体来说，我们推导出条件下的GSO恢复成本是强凸的，并使用此属性来证明，在线算法收敛到最佳的时变批量解决方案的邻域内。数值试验表明，所提出的图学习方法在适应流信息和跟踪所寻求的动态网络中的变化方面是有效的。
英文摘要：
We develop online graph learning algorithms from streaming network data. Our goal is to track the (possibly) time-varying network topology, and effect memory and computational savings by processing the data on-the-fly as they are acquired. The setup entails observations modeled as stationary graph signals generated by local diffusion dynamics on the unknown network. Moreover, we may have a priori information on the presence or absence of a few edges as in the link prediction problem. The stationarity assumption implies that the observations' covariance matrix and the so-called graph shift operator (GSO -- a matrix encoding the graph topology) commute under mild requirements. This motivates formulating the topology inference task as an inverse problem, whereby one searches for a sparse GSO that is structurally admissible and approximately commutes with the observations' empirical covariance matrix. For streaming data said covariance can be updated recursively, and we show online proximal gradient iterations can be brought to bear to efficiently track the time-varying solution of the inverse problem with quantifiable guarantees. Specifically, we derive conditions under which the GSO recovery cost is strongly convex and use this property to prove that the online algorithm converges to within a neighborhood of the optimal time-varying batch solution. Numerical tests illustrate the effectiveness of the proposed graph learning approach in adapting to streaming information and tracking changes in the sought dynamic network.
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