JOBS: Joint-Sparse Optimization from Bootstrap Samples

Abstract: Classical signal recovery based on $\ell_1$ minimization solves the least squares problem with all available measurements via sparsity-promoting regularization. In practice, it is often the case that not all measurements are available or required for recovery. Measurements might be corrupted/missing or they arrive sequentially in streaming fashion. In this paper, we propose a global sparse recovery strategy based on subsets of measurements, named JOBS, in which multiple measurements vectors are generated from the original pool of measurements via bootstrapping, and then a joint-sparse constraint is enforced to ensure support consistency among multiple predictors. The final estimate is obtained by averaging over the $K$ predictors. The performance limits associated with different choices of number of bootstrap samples $L$ and number of estimates $K$ is analyzed theoretically. Simulation results validate some of the theoretical analysis, and show that the proposed method yields state-of-the-art recovery performance, outperforming $\ell_1$ minimization and a few other existing bootstrap-based techniques in the challenging case of low levels of measurements and is preferable over other bagging-based methods in the streaming setting since it performs better with small $K$ and $L$ for data-sets with large sizes.