The recovery of missing samples from available noisy measurements is a fundamental problem in signal processing. This process is also sometimes known as graph signal inpainting, reconstruction, forecasting or inference. Many of the existing algorithms do not scale well with the size of the graph and/or they cannot be implemented efficiently in a distributed manner. In this paper, we develop efficient distributed algorithms for the recovery of time-varying graph signals. The a priori assumptions are that the signal is smooth with respect to the graph topology and correlative across time. These assumptions can be incorporated in an optimization formulation of the algorithm via Tikhonov regularization terms. Our formulation is tailored to yield algorithms that can be efficiently implemented in a distributed manner. Two different distributed algorithms, arising from two different formulations, are proposed to solve the optimization problems. The first involves the ℓ 2 -norm, and a distributed least squared recovery algorithm (DLSRA) is proposed that leverages the graph topology and sparsity of the corresponding Hessian matrix. Updates of the Hessian inverse are not required here. The second involves the ℓ 1 -norm and the philosophy of the alternating direction method of multipliers (ADMM) is utilized to develop the algorithm. An inexact Newton method is incorporated into the conventional ADMM to give a distributed ADMM recovery algorithm (DAMRA). The two distributed algorithms require only data exchanges between vertices in localized neighbourhood subgraphs. Experiments on a variety of synthetic and real-world datasets demonstrate that the proposed algorithms are superior to the existing methods in terms of the computational complexity and convergence rate.

中文翻译：

---

通过正则化问题的分布式算法恢复时变图信号


从可用的噪声测量中恢复丢失的样本是信号处理中的一个基本问题。该过程有时也称为图形信号修复、重建、预测或推理。许多现有算法不能很好地适应图的大小和/或它们不能以分布式方式有效地实现。在本文中，我们开发了有效的分布式算法来恢复时变图信号。先验假设是信号相对于图拓扑是平滑的并且与时间相关。这些假设可以通过吉洪诺夫正则化项纳入算法的优化公式中。我们的公式是专门为产生可以以分布式方式有效实现的算法而定制的。提出了由两种不同的公式产生的两种不同的分布式算法来解决优化问题。第一个涉及 ℓ 2 范数，并提出了一种分布式最小二乘恢复算法（DLSRA），该算法利用图拓扑和相应 Hessian 矩阵的稀疏性。这里不需要更新 Hessian 逆矩阵。第二个涉及 ℓ 1 范数，并利用乘子交替方向法 (ADMM) 的原理来开发算法。将不精确牛顿法融入到传统的ADMM中，给出了分布式ADMM恢复算法（DAMRA）。这两种分布式算法仅需要局部邻域子图中的顶点之间的数据交换。对各种合成和真实数据集的实验表明，所提出的算法在计算复杂度和收敛速度方面优于现有方法。