  In many applications, solutions of numerical problems are required to be
non-negative, e.g., when retrieving pixel intensity values or physical
densities of a substance. In this context, non-negative least squares (NNLS) is
a ubiquitous tool, e.g., when seeking sparse solutions of high-dimensional
statistical problems. Despite vast efforts since the seminal work of Lawson and
Hanson in the '70s, the non-negativity assumption is still an obstacle for the
theoretical analysis and scalability of many off-the-shelf solvers. In the
different context of deep neural networks, we recently started to see that the
training of overparametrized models via gradient descent leads to surprising
generalization properties and the retrieval of regularized solutions. In this
paper, we prove that, by using an overparametrized formulation, NNLS solutions
can reliably be approximated via vanilla gradient flow. We furthermore
establish stability of the method against negative perturbations of the
ground-truth. Our simulations confirm that this allows the use of vanilla
gradient descent as a novel and scalable numerical solver for NNLS. From a
conceptual point of view, our work proposes a novel approach to trading
side-constraints in optimization problems against complexity of the
optimization landscape, which does not build upon the concept of Lagrangian
multipliers.
