Cecilie Olesen Recke: Identifiability in Discrete and Continuous Graphical Lyapunov Models

Abstract.

Lyapunov models describe the steady-state distributions of linear stochastic dynamical systems. The interactions in these systems are encoded by a directed graph, which may contain cycles. In discrete time, the models arise as steady-state distributions of first-order vector autoregressive processes. In continuous time, they arise as steady-state distributions of a stochastic differential equation with linear drift. This makes them a natural framework for learning feedback structures from cross-sectional data, such as data from single-cell biology. Before this can be done in practice, however, we need to know whether the parameters are identifiable from the steady-state distribution.

When only the covariance matrix is used, identifiability holds only under restrictive assumptions. For both models, we derive tensor equations for the higher-order cumulants. Under non-Gaussianity we can use the higher-order cumulant equations to prove identifiability results using tools from algebraic geometry and combinatorics. For discrete Lyapunov models, we prove generic identifiability for all directed acyclic graphs with self-loops at every vertex and local identifiability for all directed graphs with self-loops at every vertex and no isolated vertices using cumulants up to order four. For continuous Lyapunov models, we prove generic identifiability up to scaling for all connected graphs with self-loops, using the covariance and one higher-order cumulant. Time permitting, I will discuss how such identifiability results can be used to construct estimators and to prove theoretical guarantees for them.