Sparse Graph Learning Under Laplacian-Related Constraints

TL;DR

This paper introduces a novel method for learning sparse graphs with Laplacian-related constraints using an ADMM algorithm, improving accuracy over existing methods in synthetic and real financial data.

Contribution

It proposes a modified penalized likelihood approach enforcing total positivity without requiring full Laplacian structure, along with an ADMM algorithm for optimization.

Findings

Constrained adaptive lasso outperforms existing Laplacian-based methods on synthetic data.

The approach effectively captures the underlying graph structure in real financial data.

Numerical results demonstrate significant improvements in graph learning accuracy.

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.