Graphical continuous Lyapunov models

TL;DR

This paper introduces a new graphical model class based on the linear Lyapunov equation for covariance matrices, with a structure learning method that outperforms existing algorithms and is applicable to biological network reconstruction.

Contribution

It presents a novel graphical model framework derived from Lyapunov equations and develops an $ ext{l}_1$-penalized method for structure learning, demonstrating superior performance.

Findings

The proposed method outperforms alternative algorithms in simulations.

It effectively reconstructs protein phosphorylation networks.

The model behaves predictably under marginalization.

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 $\ell_1$-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.