Sparse Graph Learning Under Laplacian-Related Constraints

Abstract

We consider the problem of learning a sparse undirected graph underlying a given set of multivariate data. We focus on graph Laplacian-related constraints on the sparse precision matrix that encodes conditional dependence between the random variables associated with the graph nodes. Under these constraints the off-diagonal elements of the precision matrix are non-positive (total positivity), and the precision matrix may not be full-rank. We investigate modifications to widely used penalized log-likelihood approaches to enforce total positivity but not the Laplacian structure. The graph Laplacian can then be extracted from the off-diagonal precision matrix. An alternating direction method of multipliers (ADMM) algorithm is presented and analyzed for constrained optimization under Laplacian-related constraints and lasso as well as adaptive lasso penalties. Numerical results based on synthetic data show that the proposed constrained adaptive lasso approach significantly outperforms existing Laplacian-based approaches. We also evaluate our approach on real financial data.

13 pages, 5 figures, 3 tables. To be published in IEEE Access

References in corpus (5)

• Sparse permutation invariant covariance estimation
• Graph Learning: A Survey
• Network exploration via the adaptive LASSO and SCAD penalties
• Robust Graph Learning from Noisy Data
• Covariance estimation with nonnegative partial correlations

Cited by in corpus (1)

• Learning Graph Laplacian with MCP