  We present a statistical framework to benchmark the performance of
reconstruction algorithms for linear inverse problems, in particular,
neural-network-based methods that require large quantities of training data. We
generate synthetic signals as realizations of sparse stochastic processes,
which makes them ideally matched to variational sparsity-promoting techniques.
We derive Gibbs sampling schemes to compute the minimum mean-square error
estimators for processes with Laplace, Student's t, and Bernoulli-Laplace
innovations. These allow our framework to provide quantitative measures of the
degree of optimality (in the mean-square-error sense) for any given
reconstruction method. We showcase our framework by benchmarking the
performance of some well-known variational methods and convolutional neural
network architectures that perform direct nonlinear reconstructions in the
context of deconvolution and Fourier sampling. Our experimental results support
the understanding that, while these neural networks outperform the variational
methods and achieve near-optimal results in many settings, their performance
deteriorates severely for signals associated with heavy-tailed distributions.
