Information Theoretically Optimal Sample Complexity of Learning Dynamical Directed Acyclic Graphs
This provides a foundational result for efficiently inferring causal structures in time-series data, with applications in fields like neuroscience and control systems.
The paper tackles the problem of learning dynamical directed acyclic graphs (DDAGs) from linear dynamical systems, deriving the optimal sample complexity as n = Θ(q log(p/q)), where p is the number of nodes and q is the maximum parents per node, with an algorithm based on power spectral density matrices.
In this article, the optimal sample complexity of learning the underlying interactions or dependencies of a Linear Dynamical System (LDS) over a Directed Acyclic Graph (DAG) is studied. We call such a DAG underlying an LDS as dynamical DAG (DDAG). In particular, we consider a DDAG where the nodal dynamics are driven by unobserved exogenous noise sources that are wide-sense stationary (WSS) in time but are mutually uncorrelated, and have the same {power spectral density (PSD)}. Inspired by the static DAG setting, a metric and an algorithm based on the PSD matrix of the observed time series are proposed to reconstruct the DDAG. It is shown that the optimal sample complexity (or length of state trajectory) needed to learn the DDAG is $n=Θ(q\log(p/q))$, where $p$ is the number of nodes and $q$ is the maximum number of parents per node. To prove the sample complexity upper bound, a concentration bound for the PSD estimation is derived, under two different sampling strategies. A matching min-max lower bound using generalized Fano's inequality also is provided, thus showing the order optimality of the proposed algorithm.