Vinzant's conjecture on 4M − 5 phase retrieval measurements
Phase retrieval with $A \in \mathbb{C}^{N \times M}$ recovers $x \in \mathbb{C}^M$, up to a global phase, from $|Ax|$; $A$ is injective if $x \bmod \mathbb{T} \mapsto |Ax|$ is injective. Bandeira, Cahill, Mixon and Nelson conjectured that $N \ge 4M - 4$ measurements are necessary and that generic $4M - 4$ measurements are injective. Conca, Edidin, Hering and Vinzant proved the second part, and Vinzant disproved the first with $11$ injective measurements in $\mathbb{C}^4$.
Problem. Let $N = 4M - 5$ and let $A \in \mathbb{C}^{N \times M}$ be random with $\operatorname{im}(A)$ uniform on the Grassmannian of $M$-dimensional subspaces of $\mathbb{C}^N$ (for example, with i.i.d. standard complex Gaussian entries). Let $p_M$ be the probability that $x \bmod \mathbb{T} \mapsto |Ax|^2$ is injective. Prove or disprove that (a) $p_M < 1$ for all $M$, and (b) $\lim_{M \to \infty} p_M = 0$.
Source: Vinzant (2015), refining a conjecture of Bandeira, Cahill, Mixon and Nelson (2014). Conjecture 19 of Randomstrasse101, in Bandeira's “Injectivity and Stability of Phase Retrieval” (2025); Entry 10 of arXiv:2603.29571.
The statement is adapted from randomstrasse101.math.ethz.ch, whose text is available under CC BY 4.0.
Classification
- Primary
- FA — Functional Analysis
- Secondary
- AG, PR
- Keywords
- frames, injectivity, phase retrieval, Randomstrasse101
- Literature checked
- Oct. 2, 2026 by clemens-admin
References
@misc{randomstrasse2025,
author = {Bandeira, Afonso S. and Dmitriev, Daniil and Lucca, Kevin and Nizi{\'c}-Nikolac, Petar and R{\"o}dder, Almut},
title = {{Randomstrasse101}: Open Problems of 2025},
year = {2026},
eprint = {2603.29571},
archivePrefix = {arXiv}
}
@article{bandeira2014saving,
author = {Bandeira, Afonso S. and Cahill, Jameson and Mixon, Dustin G. and Nelson, Aaron A.},
title = {Saving phase: Injectivity and stability for phase retrieval},
journal = {Applied and Computational Harmonic Analysis},
volume = {37},
pages = {106--125},
year = {2014}
}
@article{conca2015injectivity,
author = {Conca, Aldo and Edidin, Dan and Hering, Milena and Vinzant, Cynthia},
title = {An algebraic characterization of injectivity in phase retrieval},
journal = {Applied and Computational Harmonic Analysis},
volume = {38},
number = {2},
pages = {346--356},
year = {2015},
doi = {10.1016/j.acha.2014.06.005}
}
@inproceedings{vinzant2015small,
author = {Vinzant, Cynthia},
title = {A small frame and a certificate of its injectivity},
booktitle = {2015 International Conference on Sampling Theory and Applications (SampTA)},
pages = {197--200},
year = {2015},
doi = {10.1109/SAMPTA.2015.7148901}
}
@misc{mixon2015sampta,
author = {Mixon, Dustin G.},
title = {Conjectures from {SampTA}},
howpublished = {Short, Fat Matrices (blog)},
year = {2015},
url = {https://dustingmixon.wordpress.com/2015/07/08/conjectures-from-sampta/}
}
Resolution claims
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 …
Status history
| When | Change | By | Reason |
|---|---|---|---|
| 2026-10-09 | Open → Partially resolved | The registry's curators | On Injectivity of Phase Retrieval v1 recorded from arXiv:2606.17922v1. |
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00036, "Vinzant's conjecture on 4M − 5 phase retrieval measurements", clopen.solutions.