  The properties of gradient techniques for the phase retrieval problem have
received a considerable attention in recent years. In almost all applications,
however, the phase retrieval problem is solved using a family of algorithms
that can be interpreted as variants of Douglas-Rachford splitting. In this
work, we establish a connection between Douglas-Rachford and gradient
algorithms. Specifically, we show that in some cases a generalization of
Douglas-Rachford, called relaxed-reflect-reflect (RRR), can be viewed as
gradient descent on a certain objective function. The solutions coincide with
the critical points of that objective, which---in contrast to standard gradient
techniques---are not its minimizers. Using the objective function, we give
simple proofs of some basic properties of the RRR algorithm. Specifically, we
describe its set of solutions, show a local convexity around any solution, and
derive stability guarantees. Nevertheless, in its present state, the analysis
does not elucidate the remarkable empirical performance of RRR and its global
properties.
