---
res:
bibo_abstract:
- "In phase retrieval, we want to recover an unknown signal \U0001D465∈ℂ\U0001D451
from n quadratic measurements of the form \U0001D466\U0001D456=|⟨\U0001D44E\U0001D456,\U0001D465⟩|2+\U0001D464\U0001D456,
where \U0001D44E\U0001D456∈ℂ\U0001D451 are known sensing vectors and \U0001D464\U0001D456
is measurement noise. We ask the following weak recovery question: What is the
minimum number of measurements n needed to produce an estimator \U0001D465^(\U0001D466)
that is positively correlated with the signal \U0001D465? We consider the case
of Gaussian vectors \U0001D44E\U0001D44E\U0001D456. We prove that—in the high-dimensional
limit—a sharp phase transition takes place, and we locate the threshold in the
regime of vanishingly small noise. For \U0001D45B≤\U0001D451−\U0001D45C(\U0001D451),
no estimator can do significantly better than random and achieve a strictly positive
correlation. For \U0001D45B≥\U0001D451+\U0001D45C(\U0001D451), a simple spectral
estimator achieves a positive correlation. Surprisingly, numerical simulations
with the same spectral estimator demonstrate promising performance with realistic
sensing matrices. Spectral methods are used to initialize non-convex optimization
algorithms in phase retrieval, and our approach can boost the performance in this
setting as well. Our impossibility result is based on classical information-theoretic
arguments. The spectral algorithm computes the leading eigenvector of a weighted
empirical covariance matrix. We obtain a sharp characterization of the spectral
properties of this random matrix using tools from free probability and generalizing
a recent result by Lu and Li. Both the upper bound and lower bound generalize
beyond phase retrieval to measurements \U0001D466\U0001D456 produced according
to a generalized linear model. As a by-product of our analysis, we compare the
threshold of the proposed spectral method with that of a message passing algorithm.@eng"
bibo_authorlist:
- foaf_Person:
foaf_givenName: Marco
foaf_name: Mondelli, Marco
foaf_surname: Mondelli
foaf_workInfoHomepage: http://www.librecat.org/personId=27EB676C-8706-11E9-9510-7717E6697425
orcid: 0000-0002-3242-7020
- foaf_Person:
foaf_givenName: Andrea
foaf_name: Montanari, Andrea
foaf_surname: Montanari
bibo_doi: 10.1007/s10208-018-9395-y
bibo_issue: '3'
bibo_volume: 19
dct_date: 2019^xs_gYear
dct_isPartOf:
- http://id.crossref.org/issn/1615-3383
dct_language: eng
dct_publisher: Springer@
dct_title: Fundamental limits of weak recovery with applications to phase retrieval@
...