We 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].
