  Phase retrieval problems involve solving linear equations, but with missing
sign (or phase, for complex numbers) information. More than four decades after
it was first proposed, the seminal error reduction algorithm of (Gerchberg and
Saxton 1972) and (Fienup 1982) is still the popular choice for solving many
variants of this problem. The algorithm is based on alternating minimization;
i.e. it alternates between estimating the missing phase information, and the
candidate solution. Despite its wide usage in practice, no global convergence
guarantees for this algorithm are known. In this paper, we show that a
(resampling) variant of this approach converges geometrically to the solution
of one such problem -- finding a vector $\mathbf{x}$ from
$\mathbf{y},\mathbf{A}$, where $\mathbf{y} =
\left|\mathbf{A}^{\top}\mathbf{x}\right|$ and $|\mathbf{z}|$ denotes a vector
of element-wise magnitudes of $\mathbf{z}$ -- under the assumption that
$\mathbf{A}$ is Gaussian.
  Empirically, we demonstrate that alternating minimization performs similar to
recently proposed convex techniques for this problem (which are based on
"lifting" to a convex matrix problem) in sample complexity and robustness to
noise. However, it is much more efficient and can scale to large problems.
Analytically, for a resampling version of alternating minimization, we show
geometric convergence to the solution, and sample complexity that is off by log
factors from obvious lower bounds. We also establish close to optimal scaling
for the case when the unknown vector is sparse. Our work represents the first
theoretical guarantee for alternating minimization (albeit with resampling) for
any variant of phase retrieval problems in the non-convex setting.
