Support Recovery for the Drift Coefficient of High-Dimensional Diffusions

Consider the problem of learning the drift coefficient of a $p$ -dimensional stochastic differential equation from a sample path of length $T$ . We assume that the drift is parametrized by a high-dimensional vector, and study the support recovery problem when both $p$ and $T$ can tend to infinity. In particular, we prove a general lower bound on the sample-complexity $T$ by using a characterization of mutual information as a time integral of conditional variance, due to Kadota, Zakai, and Ziv. For linear stochastic differential equations, the drift coefficient is parametrized by a $p\times p$ matrix that describes which degrees of freedom interact under the dynamics. In this case, we analyze a $\ell_{1}$ -regularized least squares estimator and prove an upper bound on $T$ that nearly matches the lower bound on specific classes of sparse matrices.