Learning Sparse Graph with Minimax Concave Penalty under Gaussian Markov Random Fields

Tatsuya Koyakumaru, Masahiro Yukawa, Eduardo Pavez, Antonio Ortega

Research output: Contribution to journalArticlepeer-review

1 Citation (Scopus)


This paper presents a convex-analytic framework to learn sparse graphs from data. While our problem formulation is inspired by an extension of the graphical lasso using the so-called combinatorial graph Laplacian framework, a key difference is the use of a nonconvex alternative to the `1 norm to attain graphs with better interpretability. Specifically, we use the weakly-convex minimax concave penalty (the difference between the `1 norm and the Huber function) which is known to yield sparse solutions with lower estimation bias than `1 for regression problems. In our framework, the graph Laplacian is replaced in the optimization by a linear transform of the vector corresponding to its upper triangular part. Via a reformulation relying on Moreau’s decomposition, we show that overall convexity is guaranteed by introducing a quadratic function to our cost function. The problem can be solved efficiently by the primal-dual splitting method, of which the admissible conditions for provable convergence are presented. Numerical examples show that the proposed method significantly outperforms the existing graph learning methods with reasonable computation time.

Original languageEnglish
Pages (from-to)23-34
Number of pages12
JournalIEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences
Issue number1
Publication statusPublished - 2023 Jan


  • graph learning
  • graph signal processing
  • minimax concave penalty
  • primal-dual splitting method
  • proximity operator

ASJC Scopus subject areas

  • Signal Processing
  • Computer Graphics and Computer-Aided Design
  • Electrical and Electronic Engineering
  • Applied Mathematics


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