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Network estimation from multi-variate point process or time series data is aproblem of fundamental importance. Prior work has focused on parametricapproaches that require a known parametric model, which makes estimationprocedures less robust to model mis-specification, non-linearities andheterogeneities. In this paper, we develop a semi-parametric approach based onthe monotone single-index multi-variate autoregressive model (SIMAM) whichaddresses these challenges. We provide theoretical guarantees for dependentdata and an alternating projected gradient descent algorithm. Significantly wedo not explicitly assume mixing conditions on the process (although we dorequire conditions analogous to restricted strong convexity) and we achieverates of the form $O(T^{-\frac{1}{3}} \sqrt{s\log(TM)})$ (optimal in theindependent design case) where $s$ is the threshold for the maximum in-degreeof the network that indicates the sparsity level, $M$ is the number of actorsand $T$ is the number of time points. In addition, we demonstrate the superiorperformance both on simulated data and two real data examples where our SIMAMapproach out-performs state-of-the-art parametric methods both in terms ofprediction and network estimation.

Improved Prediction and Network Estimation Using the Monotone Single Index Multi-variate Autoregressive Model