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

Learning Large-Scale MTP$_2$ Gaussian Graphical Models via Bridge-Block Decomposition