  In the context of the phase retrieval problem, it is known that certain
natural classes of measurements, such as Fourier measurements and random
Bernoulli measurements, do not lead to the unique reconstruction of all
possible signals, even in combination with certain practically feasible random
masks. To avoid this difficulty, the analysis is often restricted to
measurement ensembles (or masks) that satisfy a small-ball probability
condition, in order to ensure that the reconstruction is unique.
  This paper shows a complementary result: for random Bernoulli measurements,
there is still a large class of signals that can be reconstructed uniquely,
namely those signals that are "non-peaky." In fact, this result is much more
general: it holds for random measurements sampled from any subgaussian
distribution D, without any small-ball conditions. This is demonstrated in two
ways: first, a proof of stability and uniqueness, and second, a uniform
recovery guarantee for the PhaseLift algorithm. In all of these cases, the
number of measurements m approaches the information-theoretic lower bound.
  Finally, for random Bernoulli measurements with erasures, it is shown that
PhaseLift achieves uniform recovery of all signals (including peaky ones).
