  We study an approach to solving the phase retrieval problem as it arises in a
phase-less imaging modality known as ptychography. In ptychography, small
overlapping sections of an unknown sample (or signal, say $x_0\in
\mathbb{C}^d$) are illuminated one at a time, often with a physical mask
between the sample and light source. The corresponding measurements are the
noisy magnitudes of the Fourier transform coefficients resulting from the
pointwise product of the mask and the sample. The goal is to recover the
original signal from such measurements.
  The algorithmic framework we study herein relies on first inverting a linear
system of equations to recover a fraction of the entries in $x_0 x_0^*$ and
then using non-linear techniques to recover the magnitudes and phases of the
entries of $x_0$. Thus, this paper's contributions are three-fold. First,
focusing on the linear part, it expands the theory studying which measurement
schemes (i.e., masks, shifts of the sample) yield invertible linear systems,
including an analysis of the conditioning of the resulting systems. Second, it
analyzes a class of improved magnitude recovery algorithms and, third, it
proposes and analyzes algorithms for phase recovery in the ptychographic
setting where large shifts --- up to $50\%$ the size of the mask --- are
permitted.
