  Finding sparse solutions of underdetermined linear systems commonly requires
the solving of L1 regularized least squares minimization problem, which is also
known as the basis pursuit denoising (BPDN). They are computationally expensive
since they cannot be solved analytically. An emerging technique known as deep
unrolling provided a good combination of the descriptive ability of neural
networks, explainable, and computational efficiency for BPDN. Many unrolled
neural networks for BPDN, e.g. learned iterative shrinkage thresholding
algorithm and its variants, employ shrinkage functions to prune elements with
small magnitude. Through experiments on synthetic aperture radar tomography
(TomoSAR), we discover the shrinkage step leads to unavoidable information loss
in the dynamics of networks and degrades the performance of the model. We
propose a recurrent neural network (RNN) with novel sparse minimal gated units
(SMGUs) to solve the information loss issue. The proposed RNN architecture with
SMGUs benefits from incorporating historical information into optimization, and
thus effectively preserves full information in the final output. Taking TomoSAR
inversion as an example, extensive simulations demonstrated that the proposed
RNN outperforms the state-of-the-art deep learning-based algorithm in terms of
super-resolution power as well as generalization ability. It achieved a 10% to
20% higher double scatterers detection rate and is less sensitive to phase and
amplitude ratio differences between scatterers. Test on real TerraSAR-X
spotlight images also shows a high-quality 3-D reconstruction of the test site.
