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.
Highlights
G RAPHICAL models provide a powerful tool for analyzing multivariate data [9], [23]
In a statistical graphical model, the conditional statistical dependency structure among p random variables x1, x1, · · ·, xp, (x = [x1 x2 · · · xp] ), is represented using an undirected graph where there is no edge between nodes i and j iff random variables xi and xj associated with these two nodes, are conditionally independent
The graph Laplacian can be extracted from the off-diagonal precision matrix
Summary
G RAPHICAL models provide a powerful tool for analyzing multivariate data [9], [23]. In a statistical graphical model, the conditional statistical dependency structure among p random variables x1, x1, · · · , xp, (x = [x1 x2 · · · xp] ), is represented using an undirected graph where there is no edge between nodes i and j iff random variables xi and xj associated with these two nodes, are conditionally independent. Several authors have considered Gaussian graphical models under the constraint that the distribution is multivariate totally positive of order 2 (MTP2), or equivalently, that all partial correlations are non-negative (see [24], [38] and references therein). The graph Laplacian is positive semi-definite with non-positive off-diagonal entries, can be viewed as rank-deficient precision matrix for an MTP2 Gaussian random vector Another set of approaches are based on statistical considerations under the graph Laplacian constraint [9], [10], [20], [30], [31], [40] where Laplacian L (or a generalized version) plays the role of the precision matrix Ω. A key contribution of [40] has been to show that under convex lasso ( 1) penalty, Laplacianconstrained log-likelihood approaches do not yield sparse graphs; non-convex penalties are required; see [41]
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