Quantitative stable phase retrieval in spans of independent random variables
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When does taking absolute values preserve enough information to recover a real L² function in the closed span of independent centered normalized random variables, up to its unavoidable sign? The quantitative target is stable phase retrieval.
Why it matters and what counts as progress
Why it matters: Explicit stability bounds extend qualitative recovery beyond identical distributions or finite-fourth-moment assumptions.
What counts as progress: A quantitative bound under the indexed two-sided L¹ conditions, allowing at most one exceptional coordinate, or a sharper characterization of when stability fails.
How the problem developed
R. Calderbank, I. Daubechies, D. Freeman and N. Freeman Proved stable phase retrieval in a simplified model using disjoint indicator functions attached to the random variables.
- 7 Jul 2026 · Characterization [2]
Pedro Abdalla, Jaume de Dios Pont, João P. G. Ramos and Mitchell A. Taylor Characterized stable phase retrieval in L²-spans of independent centered real random variables: after normalization, all but at most one coordinate must satisfy a uniform two-sided L¹ bound. The paper includes a second, quantitative proof that was partly LLM-generated with author guidance.
AI contribution
Stable Phase Retrieval for Spans of Independent Random Variables
Theorem 6.1 gives an explicit quantitative stable-phase-retrieval bound for the closed real L² span of independent centred normalized random variables under indexed two-sided L¹ bounds on all but at most one coordinate. Its proof uses a head–tail concentration versus diffusion argument and Sperner-type anticoncentration.
Sources
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Publication dates and editorial changes.
Model-generated strategy yields quantitative stable phase retrieval