In many specialized imaging systems, an unknown signal x C^d produces
measurements of the form
y_i = | a_i , x |2 + \eta_i , where {a_i}\subset C^d
are known measurement vectors and is an arbitrary noise term.
Because this system seems to erase the phases of the entries of x \in C^d , the
problem of reconstructing x
from y is known as the phase retrieval problem. The first approaches to this
problem were ad hoc iterative
methods which still have no theoretical guarantees on convergence. Recent
advancements including
gradient descent and convex relaxation have supplied some theoretical promises,
but often require such
conditions on the system {a_i} that they cannot be used by scientists in
practice. In particular, they tend
to require some randomness to be used in the choice of {a_i} that does not
reflect the physical systems
that typically yield the phase retrieval problem. Our work contributes a
solution to this problem which
features (a) a deterministic, practicable construction for {a_i} (b) numerical
stability with respect to noise
(c) a reconstruction algorithm with competitive runtimes. Our most recent
result is an improvement on
the robustness gained by leveraging the graph structure induced by our
measurement scheme.