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We consider the problem of estimating the parameters of a multivariateBernoulli process with auto-regressive feedback in the high-dimensional settingwhere the number of samples available is much less than the number ofparameters. This problem arises in learning interconnections of networks ofdynamical systems with spiking or binary-valued data. We allow the process todepend on its past up to a lag $p$, for a general $p \ge 1$, allowing for morerealistic modeling in many applications. We propose and analyze an$\ell_1$-regularized maximum likelihood estimator (MLE) under the assumptionthat the parameter tensor is approximately sparse. Rigorous analysis of suchestimators is made challenging by the dependent and non-Gaussian nature of theprocess as well as the presence of the nonlinearities and multi-level feedback.We derive precise upper bounds on the mean-squared estimation error in terms ofthe number of samples, dimensions of the process, the lag $p$ and other keystatistical properties of the model. The ideas presented can be used in thehigh-dimensional analysis of regularized $M$-estimators for other sparsenonlinear and non-Gaussian processes with long-range dependence.

High-Dimensional Bernoulli Autoregressive Process with Long-Range Dependence