We consider the problem of estimating the structure of an undirected weighted sparse graph underlying a set of signals, exploiting both smoothness of the signals as well as their statistics. We augment the objective function of Kalofolias (2016) which is motivated by a signal smoothness viewpoint and imposes a Laplacian constraint, with a penalized log-likelihood objective function with a lasso constraint, motivated from a statistical viewpoint. Both of these objective functions are designed for estimation of sparse graphs. An alternating direction method of multipliers (ADMM) algorithm is presented to optimize the augmented objective function. Numerical results based on synthetic data show that the proposed approach improves upon Kalofolias (2016) in estimating the inverse covariance, and improves upon graphical lasso in estimating the graph topology. We also implement an adaptive version of the proposed algorithm following adaptive lasso of Zou (2006), and empirically show that it leads to further improvement in performance.
On High-Dimensional Graph Learning Under Total Positivity
We consider the problem of estimating the structure of an undirected weighted sparse graphical model of multivariate data under the assumption that the underlying distribution is multivariate totally positive of order 2, or equivalently, all partial correlations are non-negative. Total positivity holds in several applications. The problem of Gaussian graphical model learning has been widely studied without the total positivity assumption where the problem can be formulated as estimation of the sparse precision matrix that encodes conditional dependence between random variables associated with the graph nodes. An approach that imposes total positivity is to assume that the precision matrix obeys the Laplacian constraints which include constraining the off-diagonal elements of the precision matrix to be non-positive. In this paper we investigate modifications to widely used penalized log-likelihood approaches to enforce total positivity but not the Laplacian structure. An alternating direction method of multipliers (ADMM) algorithm is presented for constrained optimization under total positivity 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, both statistical and smoothness-based non-statistical.
- Publication Date:
- NSF-PAR ID:
- Journal Name:
- 22021 55th Asilomar Conference on Signals, Systems, and Computers
- Sponsoring Org:
- National Science Foundation
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