## Abstract

The Laplacian-constrained Gaussian Markov Random Field (LGMRF) is a common multivariate statistical model for learning a weighted sparse dependency graph from given data. This graph learning problem can be formulated as a maximum likelihood estimation (MLE) of the precision matrix, subject to Laplacian structural constraints, with a sparsity-inducing penalty term. This paper aims to solve this learning problem accurately and efficiently. First, since the commonly used `_{1}-norm penalty is inappropriate in this setting and may lead to a complete graph, we employ the nonconvex minimax concave penalty (MCP), which promotes sparse solutions with lower estimation bias. Second, as opposed to existing first-order methods for this problem, we develop a second-order proximal Newton approach to obtain an efficient solver, utilizing several algorithmic features, such as using conjugate gradients, preconditioning, and splitting to active/free sets. Numerical experiments demonstrate the advantages of the proposed method in terms of both computational complexity and graph learning accuracy compared to existing methods.

Original language | American English |
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Pages (from-to) | 1171-1179 |

Number of pages | 9 |

Journal | Proceedings of Machine Learning Research |

Volume | 238 |

State | Published - 1 Jan 2024 |

Event | 27th International Conference on Artificial Intelligence and Statistics, AISTATS 2024 - Valencia, Spain Duration: 2 May 2024 → 4 May 2024 |

## All Science Journal Classification (ASJC) codes

- Artificial Intelligence
- Software
- Control and Systems Engineering
- Statistics and Probability