Consider the problem of learning the drift coefficient of a stochastic differential equation from a sample path. In this paper, we assume that the drift is parametrized by a high dimensional vector. We address the question of how long the system needs to be observed in order to learn this vector of parameters. We prove a general lower bound on this time complexity by using a characterization of mutual information as time integral of conditional variance, due to Kadota, Zakai, and Ziv. This general lower bound is applied to specific classes of linear and non-linear stochastic differential equations. In the linear case, the problem under consideration is the one of learning a matrix of interaction coefficients. We evaluate our lower bound for ensembles of sparse and dense random matrices. The resulting estimates match the qualitative behavior of upper bounds achieved by computationally efficient procedures.
SUMMARY We consider the problem of estimating sparse graphs by a lasso penalty applied to the inverse covariance matrix. Using a coordinate descent procedure for the lasso, we develop a simple algorithm—the graphical lasso—that is remarkably fast: It solves a 1000-node problem (∼500 000 parameters) in at most a minute and is 30–4000 times faster than competing methods. It also provides a conceptual link between the exact problem and the approximation suggested by Meinshausen and B¨ uhlmann (2006). We illustrate the method on some cell-signaling data from proteomics. The pattern of zero entries in the inverse covariance matrix of a multivariate normal distribution corresponds to conditional independence restrictions between variables. Covariance selection aims at estimating those structural zeros from data. We show that neighborhood selection with the Lasso is a computationally attractive alternative to standard covariance selection for sparse high-dimensional graphs. Neighborhood selection estimates the conditional independence restrictions separately for each node in the graph and is hence equivalent to variable selection for Gaussian linear models. We show that the proposed neighborhood selection scheme is consistent for sparse high-dimensional graphs. Consistency hinges on the choice of the penalty parameter. The oracle value for optimal prediction does not lead to a consistent neighborhood estimate. Controlling instead the probability of falsely joining some distinct connectivity components of the graph, consistent estimation for sparse graphs is achieved (with exponential rates), even when the number of variables grows as the number of observations raised to an arbitrary power.