On Injectivity of Phase Retrieval
This resolves a theoretical question about the injectivity threshold in phase retrieval for the benefit of researchers in signal processing and compressed sensing.
The paper proves that for a phase retrieval map generated by a matrix with i.i.d. standard complex Gaussian entries and N=4M-5, the probability of non-injectivity is positive, confirming part of a conjecture by Vinzant and Bandeira.
In this short note, we prove that if $A \in \mathbb C^{N \times M}$ with $N=4M-5$ has i.i.d.\ standard complex Gaussian entries, then the probability that the phase retrieval map generated by $A$ is not injective is positive. This proves Part (1) of a conjecture of Cynthia Vinzant, which was later restated by Afonso S. Bandeira in \cite{BDL+26}. The main result of this paper was obtained using generative AI, in particular the Rethlas system.