  We consider the phase retrieval problem for signals that belong to a union of
subspaces. We assume that amplitude measurements of the signal of length $n$
are observed after passing it through a random $m \times n$ measurement matrix.
We also assume that the signal belongs to the span of a single $d$-dimensional
subspace out of $R$ subspaces, where $d\ll n$. We assume the knowledge of all
possible subspaces, but the true subspace of the signal is unknown. We present
an algorithm that jointly estimates the phase of the measurements and the
subspace support of the signal. We discuss theoretical guarantees on the
recovery of signals and present simulation results to demonstrate the empirical
performance of our proposed algorithm. Our main result suggests that if
properly initialized, then $O(d+\log R)$ random measurements are sufficient for
phase retrieval if the unknown signal belongs to the union of $R$
low-dimensional subspaces.
