  Motivated by the learned iterative soft thresholding algorithm (LISTA), we
introduce a general class of neural networks suitable for sparse reconstruction
from few linear measurements. By allowing a wide range of degrees of
weight-sharing between the layers, we enable a unified analysis for very
different neural network types, ranging from recurrent ones to networks more
similar to standard feedforward neural networks. Based on training samples, via
empirical risk minimization we aim at learning the optimal network parameters
and thereby the optimal network that reconstructs signals from their
low-dimensional linear measurements. We derive generalization bounds by
analyzing the Rademacher complexity of hypothesis classes consisting of such
deep networks, that also take into account the thresholding parameters. We
obtain estimates of the sample complexity that essentially depend only linearly
on the number of parameters and on the depth. We apply our main result to
obtain specific generalization bounds for several practical examples, including
different algorithms for (implicit) dictionary learning, and convolutional
neural networks.
