Efficient Graph Laplacian Estimation by Proximal Newton

Yakov Medvedovsky, Eran Treister, Tirza Routtenberg

Research output: Contribution to journalConference articlepeer-review


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 languageEnglish
Pages (from-to)1171-1179
Number of pages9
JournalProceedings of Machine Learning Research
StatePublished - 1 Jan 2024
Event27th International Conference on Artificial Intelligence and Statistics, AISTATS 2024 - Valencia, Spain
Duration: 2 May 20244 May 2024

ASJC Scopus subject areas

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

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