Graphical continuous Lyapunov models

Abstract

The linear Lyapunov equation of a covariance matrix parametrizes the equilibrium covariance matrix of a stochastic process. This parametrization can be interpreted as a new graphical model class, and we show how the model class behaves under marginalization and introduce a method for structure learning via 1 -penalized loss minimization. Our proposed method is demonstrated to outperform alternative structure learning algorithms in a simulation study, and we illustrate its application for protein phosphorylation network reconstruction.

Sparse Cholesky covariance parametrization for recovering latent structure in ordered data

“…Proof. This proof follows some ideas from [17,Proposition 2.1]. By matrix calculus [30], we have the following gradient expression for j < i,…”

“…Solving Equation ( 15) can be done via proximal gradient algorithms, which have optimal convergence rates among first-order methods [25] and are tailored for a convex φ but also competitive in the non-convex case [17]. In this work we have used two such smooth loss functions: the negative Gaussian log-likelihood and the triangular Frobenious norm.…”

“…For the ordered scenario, we propose a novel learning method by directly penalising a matrix loss, in contrast with existing regression-based approaches in the literature [16]. This new estimator is feasible to compute thanks to a simplification of the loss gradient computation similar to the one proposed in [17]. Finally, we empirically assess the model performance in a broad range of experiments, using both our proposed estimators and regression-based ones.…”