  We propose and analyze a continuous-time firing-rate neural network, the
positive firing-rate competitive network (\pfcn), to tackle sparse
reconstruction problems with non-negativity constraints. These problems, which
involve approximating a given input stimulus from a dictionary using a set of
sparse (active) neurons, play a key role in a wide range of domains, including
for example neuroscience, signal processing, and machine learning. First, by
leveraging the theory of proximal operators, we relate the equilibria of a
family of continuous-time firing-rate neural networks to the optimal solutions
of sparse reconstruction problems. Then, we prove that the \pfcn is a positive
system and give rigorous conditions for the convergence to the equilibrium.
Specifically, we show that the convergence: (i) only depends on a property of
the dictionary; (ii) is linear-exponential, in the sense that initially the
convergence rate is at worst linear and then, after a transient, it becomes
exponential. We also prove a number of technical results to assess the
contractivity properties of the neural dynamics of interest. Our analysis
leverages contraction theory to characterize the behavior of a family of
firing-rate competitive networks for sparse reconstruction with and without
non-negativity constraints. Finally, we validate the effectiveness of our
approach via a numerical example.
