A note on Douglas-Rachford, subgradients, and phase retrieval
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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 subgradient 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 analyze the RRR algorithm, 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.
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