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 × p matrix that describes which degrees of freedom interact under the dynamics. In this case, we analyze a ℓ₁-regularized least squares estimator and prove an upper bound on T that nearly matches the lower bound on specific classes of sparse matrices.