  This paper studies the problem of learning the large-scale Gaussian graphical
models that are multivariate totally positive of order two ($\text{MTP}_2$). By
introducing the concept of bridge, which commonly exists in large-scale sparse
graphs, we show that the entire problem can be equivalently optimized through
(1) several smaller-scaled sub-problems induced by a \emph{bridge-block
decomposition} on the thresholded sample covariance graph and (2) a set of
explicit solutions on entries corresponding to bridges. From practical aspect,
this simple and provable discipline can be applied to break down a large
problem into small tractable ones, leading to enormous reduction on the
computational complexity and substantial improvements for all existing
algorithms. The synthetic and real-world experiments demonstrate that our
proposed method presents a significant speed-up compared to the
state-of-the-art benchmarks.
