We propose two new least mean squares (LMS)-based algorithms for adaptive estimation of graph signals that improve the convergence speed of the LMS algorithm while preserving its low computational complexity. The first algorithm, named extended least mean squares (ELMS), extends the LMS algorithm by virtue of reusing the signal vectors of previous iterations alongside the signal available at the current iteration. Utilizing the previous signal vectors accelerates the convergence of the ELMS algorithm at the expense of higher steady-state error compared to the LMS algorithm. To further improve the performance, we propose the fast ELMS (FELMS) algorithm in which the influence of the signal vectors of previous iterations is controlled by optimizing the gradient of the mean-square deviation (GMSD). The FELMS algorithm converges faster than the ELMS algorithm and has steady-state errors comparable to that of the LMS algorithm. We analyze the mean-square performance of ELMS and FELMS algorithms theoretically and derive the respective convergence conditions as well as the predicted MSD values. In addition, we present an adaptive sampling strategy in which the sampling probability of each node is changed according to the GMSD of the node. Computer simulations using both synthetic and real data validate the theoretical results and demonstrate the merits of the proposed algorithms.

中文翻译：

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使用自适应采样有效估计图信号

我们提出了两种新的基于最小均方 (LMS) 的算法用于图信号的自适应估计，这些算法提高了 LMS 算法的收敛速度，同时保持其低计算复杂度。第一种算法称为扩展最小均方 (ELMS)，通过重用先前迭代的信号向量以及当前迭代可用的信号来扩展 LMS 算法。与 LMS 算法相比，利用先前的信号向量以更高的稳态误差为代价加速了 ELMS 算法的收敛。为了进一步提高性能，我们提出了快速ELMS（FELMS）算法，其中通过优化均方偏差（GMSD）的梯度来控制先前迭代信号向量的影响。FELMS 算法的收敛速度比 ELMS 算法快，并且具有与 LMS 算法相当的稳态误差。我们从理论上分析了 ELMS 和 FELMS 算法的均方性能，并推导出各自的收敛条件以及预测的 MSD 值。此外，我们提出了一种自适应采样策略，其中每个节点的采样概率根据节点的 GMSD 变化。使用合成数据和真实数据的计算机模拟验证了理论结果并证明了所提出算法的优点。我们提出了一种自适应采样策略，其中每个节点的采样概率根据节点的 GMSD 变化。使用合成数据和真实数据的计算机模拟验证了理论结果并证明了所提出算法的优点。我们提出了一种自适应采样策略，其中每个节点的采样概率根据节点的 GMSD 变化。使用合成数据和真实数据的计算机模拟验证了理论结果并证明了所提出算法的优点。