  The iterations of many first-order algorithms, when applied to minimizing
common regularized regression functions, often resemble neural network layers
with pre-specified weights. This observation has prompted the development of
learning-based approaches that purport to replace these iterations with
enhanced surrogates forged as DNN models from available training data. For
example, important NP-hard sparse estimation problems have recently benefitted
from this genre of upgrade, with simple feedforward or recurrent networks
ousting proximal gradient-based iterations. Analogously, this paper
demonstrates that more powerful Bayesian algorithms for promoting sparsity,
which rely on complex multi-loop majorization-minimization techniques, mirror
the structure of more sophisticated long short-term memory (LSTM) networks, or
alternative gated feedback networks previously designed for sequence
prediction. As part of this development, we examine the parallels between
latent variable trajectories operating across multiple time-scales during
optimization, and the activations within deep network structures designed to
adaptively model such characteristic sequences. The resulting insights lead to
a novel sparse estimation system that, when granted training data, can estimate
optimal solutions efficiently in regimes where other algorithms fail, including
practical direction-of-arrival (DOA) and 3D geometry recovery problems. The
underlying principles we expose are also suggestive of a learning process for a
richer class of multi-loop algorithms in other domains.
