  This work introduces a highly scalable spectral graph densification framework
for learning resistor networks with linear measurements, such as node voltages
and currents. We prove that given $O(\log N)$ pairs of voltage and current
measurements, it is possible to recover ultra-sparse $N$-node resistor networks
which can well preserve the effective resistance distances on the graph. Also,
the learned graphs preserve the structural (spectral) properties of the
original graph, which can potentially be leveraged in many circuit design and
optimization tasks. We show that the proposed graph learning approach is
equivalent to solving the classical graphical Lasso problems with
Laplacian-like precision matrices. Through extensive experiments for a variety
of real-world test cases, we show that the proposed approach is highly scalable
for learning ultra-sparse resistor networks without sacrificing solution
quality.
