  We propose a new algorithm to learn a dictionary for reconstructing and
sparsely encoding signals from measurements without phase. Specifically, we
consider the task of estimating a two-dimensional image from squared-magnitude
measurements of a complex-valued linear transformation of the original image.
Several recent phase retrieval algorithms exploit underlying sparsity of the
unknown signal in order to improve recovery performance. In this work, we
consider such a sparse signal prior in the context of phase retrieval, when the
sparsifying dictionary is not known in advance. Our algorithm jointly
reconstructs the unknown signal - possibly corrupted by noise - and learns a
dictionary such that each patch of the estimated image can be sparsely
represented. Numerical experiments demonstrate that our approach can obtain
significantly better reconstructions for phase retrieval problems with noise
than methods that cannot exploit such "hidden" sparsity. Moreover, on the
theoretical side, we provide a convergence result for our method.
