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Abstract
We consider the problem of estimating the parameters of a linear autoregressive model with sub-Gaussian innovations from a limited sequence of consecutive observations. Assuming that the parameters are compressible, we analyze the performance of the ℓ1-regularized least squares as well as a greedy estimator of the parameters and characterize the sampling trade-offs required for stable recovery in the non-asymptotic regime. Our results extend those of compressed sensing for linear models where the covariates are i.i.d. and independent of the observation history to autoregressive processes with highly inter-dependent covariates. We also derive sufficient conditions on the sparsity level that guarantee the minimax optimality of the ℓ1-regularized least squares estimate. Applying these techniques to simulated data as well as real-world datasets from crude oil prices and traffic speed data confirm our predicted theoretical performance gains in terms of estimation accuracy and model selection.