Distributed Batch Reconstruction of Time-varying Graph Signals via Sobolev Smoothness on Cartesian Product Graph
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摘要: 针对大规模网络数据的重构问题，该文以图信号处理(GSP)理论为基础，提出一种基于笛卡尔乘积图上Sobolev平滑的分布式批量重构算法(DBR-SSC)。该算法首先按时间顺序将时变图信号划分为多个信号段，并利用笛卡尔积将每一段内各时刻的图建模为乘积图；然后利用笛卡尔乘积图上的Sobolev差分平滑，将每一段的信号重构问题归结为优化问题；最后设计具有高收敛速度的分布式算法求解该优化问题。采用两种现实世界的数据集进行仿真实验，实验结果表明所提算法重构误差低并具有高收敛速度。Abstract: For the reconstruction of large-scale network data, a Distributed Batch Reconstruction algorithm via Sobolev Smoothness on Cartesian product graph (DBR-SSC) is proposed, which is based on the Graph Signal Processing (GSP) theory. In the proposed algorithm, the time-varying graph signal is firstly divided into multiple signal segments in time dimension, and a product graph is constructed from graphs at each time instant via Cartesian product. Secondly, the reconstruction of the time-varying graph signal in each segment is formulated as an optimization problem by exploiting the Sobolev difference smoothness on the Cartesian product graph. Finally, a distributed algorithm with high convergence rate is devised to solve the optimization problem. Two real world data sets are used for experiments, and it is shown that the proposed algorithm has low reconstruction error and high convergence rate.
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算法1 段内时变图信号$ \bar {\boldsymbol{x}} $的分布式批量重构算法 输入：矩阵$\bar {\boldsymbol{H}}$，观测信号$\bar {\boldsymbol{y}}$，迭代停止标准$ \varepsilon $，最大迭代次数$ K $, ${\bar {\boldsymbol{x}}^{(0)} }{\text{ = } }{{ {\textit{0}}}}$, $ t = 0 $。 步骤： 步骤1 根据式(12)计算$\bar {\boldsymbol{P}}$。 步骤2 每一个子图中心节点(m,i)计算 ${{\boldsymbol{\chi}} _{(m,i)} }{\bar {\boldsymbol{x} }^{(t + 1)} } = {{\boldsymbol{\chi}} _{(m,i)} }{\bar {\boldsymbol{x} }^{(t)} } - {{\boldsymbol{\chi}} _{(m,i)} }\bar {\boldsymbol{P} }{{\boldsymbol{\chi}} _{(m,i)} }({{\boldsymbol{\chi}} _{(m,i)} }\bar {\boldsymbol{H} }{{\boldsymbol{\chi}} _{(m,i)} }{\bar {\boldsymbol{x} }^{(t)} } - {{\boldsymbol{\chi}} _{(m,i)} }\bar {\boldsymbol{y} })$ (14) 并向${\bar {\boldsymbol{v}}_{(m,i)} }$中的所有节点$ (k,l) $发送相应的信号值${{\boldsymbol{\chi}} _{(m,i)} }{\bar {\boldsymbol{x} }^{(t{\text{ + } }1)} }{(h)_{h = N \times (k - 1) + l} }$。 步骤3 $ \bar V $中的各节点对接收到的信号值进行平均，然后将平均值分别发送给所在子图的中心节点。 步骤4 各子图中心节点(m,i)将收到的信号值以及自身的信号值组成新的子图信号${{\boldsymbol{\chi}} _{(m,i)} }{\bar {\boldsymbol{x} }^{(t{\text{ + } }1)} }$，如果
$\left\| { {{\boldsymbol{\chi}} _{(m,i)} }{ {\bar {\boldsymbol{x} } }^{(t + 1)} } - {{\boldsymbol{\chi}} _{(m,i)} }{ {\bar {\boldsymbol{x} } }^{(t)} } } \right\|_2^2 < \varepsilon$成立或$ t > K $，则结束程序。否则，令$ t = t + 1 $，并返回步骤2。输出：${\bar {\boldsymbol{x}}^{(t + 1)} }$作为本段的重构信号。 
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表 1 PM2.5数据集上各种算法的平均重构误差(RMSE)对比
算法 受损节点数量占比(%) 10 20 30 BR-DS 1.29 1.94 2.43 BR-SS 1.29 1.94 2.43 BR-TR 1.60 2.37 2.91 DBR-SSC 1.29 1.91 2.41
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表 2 PM2.5数据集上各种算法的平均迭代次数对比
算法 受损节点数量占比(%) 10 20 30 BR-DS 1.56 2.72 3.52 BR-SS 1.53 2.54 3.28 BR-TR 0.43 0.60 0.66 DBR-SSC 0.15 0.16 0.20
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表 3 PM2.5数据集上各种算法所用平均CPU时间对比 (10–3 s)
算法 受损节点数量占比(%) 10 20 30 BR-DS 1.18 1.95 2.45 BR-SS 1.17 1.89 2.32 BR-TR 0.39 0.45 0.49 DBR-SSC 2.01 2.01 2.03
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表 4 COVID-19数据集上各种算法的平均重构误差(RMSE)对比
算法 受损节点数量占比(%) 10 20 30 BR-DS 204 341 463 BR-SS 204 341 462 BR-TR 581 996 1254 DBR-SSC 196 332 447
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表 5 COVID-19数据集上各种算法的平均迭代次数对比
算法 受损节点数量占比(%) 10 20 30 BR-DS 6.64 6.67 6.67 BR-SS 4.27 6.64 6.67 BR-TR 0.56 0.70 0.85 DBR-SSC 0.33 0.54 0.75
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表 6 COVID-19数据集上各种算法所用平均CPU时间对比 (10–3 s)
算法 受损节点数量占比(%) 10 20 30 BR-DS 12.56 12.56 12.67 BR-SS 8.44 12.63 12.59 BR-TR 1.20 1.41 1.73 DBR-SSC 7.97 8.12 8.19
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