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

• URL: http://arxiv.org/abs/2309.13405v3
• Date: Fri, 29 Sep 2023 12:50:55 GMT
• Title: Learning Large-Scale MTP$_2$ Gaussian Graphical Models via Bridge-Block Decomposition
• Authors: Xiwen Wang, Jiaxi Ying, Daniel P. Palomar
• Abstract summary: We show that the entire problem can be equivalently optimized through several smaller-scaled sub-problems. From practical aspect, this simple and provable discipline can be applied to break down a large problem into small tractable ones.
• Score: 15.322124183968635
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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.

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