  We study the low-rank phase retrieval problem, where we try to recover a
$d_1\times d_2$ low-rank matrix from a series of phaseless linear measurements.
This is a fourth-order inverse problem, as we are trying to recover factors of
matrix that have been put through a quadratic nonlinearity after being
multiplied together.
  We propose a solution to this problem using the recently introduced technique
of anchored regression. This approach uses two different types of convex
relaxations: we replace the quadratic equality constraints for the phaseless
measurements by a search over a polytope, and enforce the rank constraint
through nuclear norm regularization. The result is a convex program that works
in the space of $d_1 \times d_2$ matrices.
  We analyze two specific scenarios. In the first, the target matrix is
rank-$1$, and the observations are structured to correspond to a phaseless
blind deconvolution. In the second, the target matrix has general rank, and we
observe the magnitudes of the inner products against a series of independent
Gaussian random matrices. In each of these problems, we show that the anchored
regression returns an accurate estimate from a near-optimal number of
measurements given that we have access to an anchor matrix of sufficient
quality. We also show how to create such an anchor in the phaseless blind
deconvolution problem, again from an optimal number of measurements, and
present a partial result in this direction for the general rank problem.
