Causal Structure Recovery of Linear Dynamical Systems: An FFT based Approach

Learning causal effects from data is a fundamental and well-studied problem across science, especially when the cause-effect relationship is static in nature. However, causal effect is less explored when there are dynamical dependencies, i.e., when dependencies exist between entities across time. Identifying dynamic causal effects from time-series observations is computationally expensive when compared to the static scenario. We demonstrate that the computational complexity of recovering the causation structure for the vector auto-regressive (VAR) model is $O(Tn^3N^2)$, where $n$ is the number of nodes, $T$ is the number of samples, and $N$ is the largest time-lag in the dependency between entities. We report a method, with a reduced complexity of $O(Tn^3 \log N)$, to recover the causation structure to obtain frequency-domain (FD) representations of time-series. Since FFT accumulates all the time dependencies on every frequency, causal inference can be performed efficiently by considering the state variables as random variables at any given frequency. We additionally show that, for systems with interactions that are LTI, do-calculus machinery can be realized in the FD resulting in versions of the classical single-door (with cycles), front and backdoor criteria. We demonstrate, for a large class of problems, graph reconstruction using multivariate Wiener projections results in a significant computational advantage with $O(n)$ complexity over reconstruction algorithms such as the PC algorithm which has $O(n^q)$ complexity, where $q$ is the maximum neighborhood size. This advantage accrues due to some remarkable properties of the phase response of the frequency-dependent Wiener coefficients which is not present in any time-domain approach.

Information Theoretically Optimal Sample Complexity of Learning Dynamical Directed Acyclic Graphs

In this article, the optimal sample complexity of learning the underlying interaction/dependencies of a Linear Dynamical System (LDS) over a Directed Acyclic Graph (DAG) is studied. The sample complexity of learning a DAG's structure is well-studied for static systems, where the samples of nodal states are independent and identically distributed (i.i.d.). However, such a study is less explored for DAGs with dynamical systems, where the nodal states are temporally correlated. We call such a DAG underlying an LDS as \emph{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 settings, a metric and an algorithm based on the PSD matrix of the observed time series are proposed to reconstruct the DDAG. The equal noise PSD assumption can be relaxed such that identifiability conditions for DDAG reconstruction are not violated. For the LDS with WSS (sub) Gaussian exogenous noise sources, it is shown that the optimal sample complexity (or length of state trajectory) needed to learn the DDAG is $n=\Theta(q\log(p/q))$, 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.