High-Dimensional Bernoulli Autoregressive Process with Long-Range Dependence

TL;DR

This paper develops a high-dimensional estimation method for multivariate Bernoulli autoregressive processes with long-range dependence, providing theoretical error bounds and addressing challenges from dependence and nonlinearity.

Contribution

It introduces an $ ext{l}_1$-regularized MLE for sparse parameters in high-dimensional Bernoulli autoregressive models with long-range dependence, with rigorous error analysis.

Findings

Derived upper bounds on estimation error based on sample size and model parameters.

Extended analysis techniques to dependent, non-Gaussian, nonlinear processes with long-range dependence.

Applicable to network models with binary or spiking data.

Abstract

We consider the problem of estimating the parameters of a multivariate Bernoulli process with auto-regressive feedback in the high-dimensional setting where the number of samples available is much less than the number of parameters. This problem arises in learning interconnections of networks of dynamical systems with spiking or binary-valued data. We allow the process to depend on its past up to a lag $p$, for a general $p \ge 1$, allowing for more realistic modeling in many applications. We propose and analyze an $\ell_1$-regularized maximum likelihood estimator (MLE) under the assumption that the parameter tensor is approximately sparse. Rigorous analysis of such estimators is made challenging by the dependent and non-Gaussian nature of the process 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 of the number of samples, dimensions of the process, the lag $p$ and other key statistical properties of the model. The ideas presented can be used in the high-dimensional analysis of regularized $M$-estimators for other sparse nonlinear and non-Gaussian processes with long-range dependence.