The Minimum Number of Measurements for Almost-Everywhere Complex Phase Retrieval

What the math gives to ML

The paper establishes an exact information-theoretic threshold for recovering a generic complex vector when only phase-invariant intensity measurements are retained: 2d real measurements are necessary and generically sufficient, while 2d-1 cannot be almost-everywhere injective modulo global phase. This is useful for complex-valued neural representations that intentionally discard global phase, including magnitude-only latent bottlenecks and optical encoders. The practical transfer is to size such a bottleneck at exactly 2d learned intensity channels and compare it against the provably insufficient 2d-1 design, rather than selecting the width heuristically.