Gaussian Critical-Threshold Instability in Real Phase Retrieval
Abstract
We establish the natural scale of instability in real Gaussian phase retrieval at the critical injectivity threshold. Let A be a (2M−1)× M matrix with independent standard Gaussian entries and let KM=binom2M−1M. For every wM→∞ with log wM=o(M), we prove that ℙ{(wM√MKM)−1≤ω(A)≤ wM/(√MKM)}→1, where ω(A) is the Balan--Wang stability parameter. Consequently, −M−1logω(A)→log4 in probability. For full-spark matrices at this threshold, ω(A) equals both the optimal lower Lipschitz constant of x↦|Ax| and the minimum least singular value over all square row submatrices. The upper bound follows from a second-moment analysis of overlapping minors. A weighted Gaussian inverse-tail asymptotic and inverse-Wishart concentration yield asymptotic independence for central overlaps; rectangular hard-edge bounds control the remaining overlaps. The matching lower bound follows from a union bound and a square Gaussian hard-edge estimate.
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BibTeX
@misc{haggblom2026gaussian,
title = {{Gaussian Critical-Threshold Instability in Real Phase Retrieval}},
author = {Christian E. Häggblom},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.28198},
eprint = {2609.28198},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.28198},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Gaussian Critical-Threshold Instability in Real Phase Retrieval
AU - Christian E. Häggblom
PY - 2026
DA - 2026/09/23
PB - arXiv
DO - 10.48550/arxiv.2609.28198
UR - https://arxiv.org/abs/2609.28198
AB - We establish the natural scale of instability in real Gaussian phase retrieval at the critical injectivity threshold. Let A be a (2M−1)× M matrix with independent standard Gaussian entries and let K_M=binom2M−1M. For every w_M→∞ with log w_M=o(M), we prove that ℙ{(w_M√MK_M)^−1≤ω(A)≤ w_M/(√MK_M)}→1, where ω(A) is the Balan--Wang stability parameter. Consequently, −M^−1logω(A)→log4 in probability. For full-spark matrices at this threshold, ω(A) equals both the optimal lower Lipschitz constant of x↦|Ax| and the minimum least singular value over all square row submatrices. The upper bound follows from a second-moment analysis of overlapping minors. A weighted Gaussian inverse-tail asymptotic and inverse-Wishart concentration yield asymptotic independence for central overlaps; rectangular hard-edge bounds control the remaining overlaps. The matching lower bound follows from a union bound and a square Gaussian hard-edge estimate.
ER -