Injectivity thresholds for complex phase retrieval
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For M≥2 and N=4M−5, can an open set of complex N×M measurement matrices fail injective phase retrieval, which recovers a vector up to global phase from measurement magnitudes? Equivalently, the refined random-matrix target asks whether Gaussian measurements fail with positive probability.
Why it matters and what counts as progress
Why it matters: Open noninjective measurement families establish a positive failure probability, which an isolated exceptional matrix would not show.
What counts as progress: Prove an open noninjective family in every M or determine complementary injectivity probabilities and measurement thresholds.
How the problem developed
- 2014 · Conjecture [1]
Bandeira, Cahill, Mixon and Nelson They conjectured that 4M−4 generic complex measurements suffice and that this number is necessary for injectivity.
- 2015 · Counterexample [2]
Cynthia Vinzant Vinzant exhibited 11 injective measurements in C^4, disproving the conjectured necessity and motivating a refined random-frame question at N=4M−5.
- 16 Jun 2026 · Reported partial result [3]
Zhangsong Li The preprint proves that for every M≥2 with N=4M−5, a nonempty open set of complex measurement matrices is noninjective; it settles p_M<1 but not the separate limit p_M→0.
AI contribution
On Injectivity of Phase Retrieval
For every M≥2 with N=4M−5, a nonempty open set of complex N×M measurement matrices has a noninjective phase-retrieval map. Hence an IID complex Gaussian matrix is noninjective with positive probability, so p_M<1.
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