  The iterations of many sparse estimation algorithms are comprised of a fixed
linear filter cascaded with a thresholding nonlinearity, which collectively
resemble a typical neural network layer. Consequently, a lengthy sequence of
algorithm iterations can be viewed as a deep network with shared, hand-crafted
layer weights. It is therefore quite natural to examine the degree to which a
learned network model might act as a viable surrogate for traditional sparse
estimation in domains where ample training data is available. While the
possibility of a reduced computational budget is readily apparent when a
ceiling is imposed on the number of layers, our work primarily focuses on
estimation accuracy. In particular, it is well-known that when a signal
dictionary has coherent columns, as quantified by a large RIP constant, then
most tractable iterative algorithms are unable to find maximally sparse
representations. In contrast, we demonstrate both theoretically and empirically
the potential for a trained deep network to recover minimal $\ell_0$-norm
representations in regimes where existing methods fail. The resulting system is
deployed on a practical photometric stereo estimation problem, where the goal
is to remove sparse outliers that can disrupt the estimation of surface normals
from a 3D scene.
