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Large-scale recurrent networks have drawn increasing attention recently because of their capabilities in modeling a large variety of real-world phenomena and physical mechanisms. This paper studies how to identify all authentic connections and estimate system parameters of a recurrent network, given a sequence of node observations. This task becomes extremely challenging in modern network applications, because the available observations are usually very noisy and limited, and the associated dynamical system is strongly nonlinear. By formulating the problem as multivariate sparse sigmoidal regression, we develop simple-to-implement network learning algorithms, with rigorous convergence guarantee in theory, for a variety of sparsity-promoting penalty forms. A quantile variant of progressive recurrent network screening is proposed for efficient computation and allows for direct cardinality control of network topology in estimation. Moreover, we investigate recurrent network stability conditions in Lyapunov's sense, and integrate such stability constraints into sparse network learning. Experiments show excellent performance of the proposed algorithms in network topology identification and forecasting. [ABSTRACT FROM PUBLISHER]

We introduce and study the group square-root lasso (GSRL) method for estimation in high dimensional sparse regression models with group structure. The new estimator minimizes the square root of the residual sum of squares plus a penalty term proportional to the sum of the Euclidean norms of groups of the regression parameter vector. The net advantage of the method over the existing group lasso-type procedures consists in the form of the proportionality factor used in the penalty term, which for GSRL is independent of the variance of the error terms. This is of crucial importance in models with more parameters than the sample size, when estimating the variance of the noise becomes as difficult as the original problem. We show that the GSRL estimator adapts to the unknown sparsity of the regression vector, and has the same optimal estimation and prediction accuracy as the GL estimators, under the same minimal conditions on the model. This extends the results recently established for the square-root lasso, for sparse regression without group structure. Moreover, as a new type of result for square-root lasso methods, with or without groups, we study correct pattern recovery, and show that it can be achieved under conditions similar to those needed by the lasso or group-lasso-type methods, but with a simplified tuning strategy. We implement our method via a new algorithm, with proved convergence properties, which, unlike existing methods, scales well with the dimension of the problem. Our simulation studies support strongly our theoretical findings. [ABSTRACT FROM AUTHOR]