This work introduces a highly-scalable spectral graph densification framework (SGL) for learning resistor networks with linear measurements, such as node voltages and currents. We show that the proposed graph learning approach is equivalent to solving the classical graphical Lasso problems with Laplacian-like precision matrices. We prove that given $O(\log N)$ pairs of voltage and current measurements, it is possible to recover sparse $N$-node resistor networks that can well preserve the effective resistance distances on the original graph. In addition, the learned graphs also preserve the structural (spectral) properties of the original graph, which can potentially be leveraged in many circuit design and optimization tasks. To achieve more scalable performance, we also introduce a solver-free method (SF-SGL) that exploits multilevel spectral approximation of the graphs and allows for a scalable and flexible decomposition of the entire graph spectrum (to be learned) into multiple different eigenvalue clusters (frequency bands). Such a solver-free approach allows us to more efficiently identify the most spectrally-critical edges for reducing various ranges of spectral embedding distortions. Through extensive experiments for a variety of real-world test cases, we show that the proposed approach is highly scalable for learning sparse resistor networks without sacrificing solution quality. We also introduce a data-driven EDA algorithm for vectorless power/thermal integrity verifications to allow estimating worst-case voltage/temperature (gradient) distributions across the entire chip by leveraging a few voltage/temperature measurements.

翻译：本文提出了一种高可扩展性的谱图稠密化框架（SGL），用于学习具有线性测量（如节点电压和电流）的电阻网络。我们证明所提出的图学习方法等价于求解具有拉普拉斯型精度矩阵的经典图形套索问题。我们证明，给定$O(\log N)$对电压和电流测量值，可以恢复稀疏的$N$节点电阻网络，且该网络能较好地保持原始图上的有效电阻距离。此外，学习得到的图还保持了原始图的结构（谱）特性，这有望被应用于众多电路设计与优化任务中。为了实现更具可扩展性的性能，我们还提出了一种免求解器方法（SF-SGL），该方法利用图的多级谱近似，并允许将整个图谱（待学习）可扩展且灵活地分解为多个不同的特征值簇（频带）。这种免求解器方法使我们能更高效地识别最具谱关键性的边，以减少各种谱嵌入失真的范围。通过对多种真实世界测试案例的大量实验，我们证明了所提出的方法在不牺牲解质量的前提下，在学习稀疏电阻网络方面具有高度可扩展性。我们还引入了一种数据驱动的电子设计自动化算法，用于无向量功耗/热完整性验证，通过利用少量电压/温度测量值，可估计整个芯片上的最差情况电压/温度（梯度）分布。