Phase retrieval without small-ball probability assumptions: Stability and uniqueness

We study stability and uniqueness for the phase retrieval problem. That is, we ask when is a signal x ε Rⁿ stably and uniquely determined (up to small perturbations), when one performs phaseless measurements of the form y_i = a^T_ix² (for i = 1..., N), where the vectors a_i ε Rⁿ are chosen independently at random, with each coordinate a_ij ε 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 ⊂ Rⁿ 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 ε T depends on the complexity of T in a natural way, extending previous results of Eldar and Mendelson [12].