



01770nas a2200145   4500008004100000245006300041210006200104260001500166300001200181490000700193520134000200100001901540700001601559856004901575       2018                        eng d00aPhase Retrieval Without Small-Ball Probability Assumptions0 aPhase Retrieval Without SmallBall Probability Assumptions  c2018/01/01  a485-5000 v643 a<p><span style="color: rgb(51, 51, 51); font-family: sans-serif; font-size: 15px;">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 2), without any small-ball conditions. This is demonstrated in two ways: 1) a proof of stability and uniqueness and 2) 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).</span></p>1 aKrahmer, Felix1 aLiu, Yi-Kai  uhttp://ieeexplore.ieee.org/document/8052535/01564nas a2200133   4500008004100000245008900041210006900130260001500199300001200214520109500226100001901321700001601340856007401356       2015                        eng d00aPhase Retrieval Without Small-Ball Probability Assumptions: Stability and Uniqueness0 aPhase Retrieval Without SmallBall Probability Assumptions Stabil  c2015/05/25  a411-4143 aWe study stability and uniqueness for the phase
retrieval problem. That is, we ask when is a signal x ∈ R
n
stably and uniquely determined (up to small perturbations), when
one performs phaseless measurements of the form yi = |a
T
i x|
2
(for i = 1, . . . , N), where the vectors ai ∈ R
n
are chosen
independently at random, with each coordinate aij ∈ R being
chosen independently from a fixed sub-Gaussian distribution
D. It is well known that for many common choices of D,
certain ambiguities can arise that prevent x from being uniquely
determined.
In this note we show that for any sub-Gaussian distribution
D, with no additional assumptions, most vectors x cannot lead
to such ambiguities. More precisely, we show stability and
uniqueness for all sets of vectors T ⊂ R
n which are not
too peaky, in the sense that at most a constant fraction of
their mass is concentrated on any one coordinate. The number
of measurements needed to recover x ∈ T depends on the
complexity of T in a natural way, extending previous results
of Eldar and Mendelson [12].1 aKrahmer, Felix1 aLiu, Yi-Kai  uhttp://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=7148923&tag=1