Sparse Graph Learning Under Laplacian-Related Constraints
This addresses graph learning for applications like financial data analysis, but appears incremental as it modifies existing penalized log-likelihood approaches.
The paper tackles the problem of learning sparse undirected graphs from multivariate data by imposing Laplacian-related constraints on the precision matrix, proposing a constrained adaptive lasso approach with an ADMM algorithm. Numerical results on synthetic data show it significantly outperforms existing Laplacian-based methods.
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.