In this article, we are interested in adaptive and distributed estimation of graph filters from streaming data. We formulate this problem as a consensus estimation problem over graphs, which can be addressed with diffusion LMS strategies. Most popular graph-shift operators such as those based on the graph Laplacian matrix, or the adjacency matrix, are not energy preserving. This may result in an ill-conditioned estimation problem, and reduce the convergence speed of the distributed algorithms. To address this issue and improve the transient performance, we introduce a preconditioned graph diffusion LMS algorithm. We also propose a computationally efficient version of this algorithm by approximating the Hessian matrix with local information. Performance analyses in the mean and mean-square sense are provided. Finally, we consider a more general problem where the filter coefficients to estimate may vary over the graph. To avoid a large estimation bias, we introduce an unsupervised clustering method for splitting the global estimation problem into local ones. Numerical results show the effectiveness of the proposed algorithms and validate the theoretical results.

中文翻译：

---

具有多任务图过滤器模型的图的在线分布式学习

在本文中，我们对根据流数据自适应和分布式估计图滤波器感兴趣。我们将此问题表述为图上的共识估计问题，可以通过扩散LMS策略解决。最流行的图移算子，例如基于图拉普拉斯矩阵或邻接矩阵的算子，并不是节能的。这可能会导致状况不佳的估计问题，并降低分布式算法的收敛速度。为了解决此问题并提高暂态性能，我们引入了预处理图扩散LMS算法。我们还通过用局部信息近似Hessian矩阵来提出该算法的计算有效版本。提供了均值和均方意义上的性能分析。最后，我们考虑一个更普遍的问题，其中要估计的滤波器系数可能会在整个图表上变化。为了避免较大的估计偏差，我们引入了一种无监督的聚类方法，将全局估计问题分为局部问题。数值结果表明了所提算法的有效性，并验证了理论结果。