Fundamental Limits of Weak Recovery with Applications to Phase Retrieval
In phase retrieval, we want to recover an unknown signal x ∈ C d from n quadratic measurements of the form y i = | ⟨ a i , x ⟩ | 2 + w i , where a i ∈ C d are known sensing vectors and w i is measurement noise. We ask the following weak recovery question: What is the minimum number of measurements n...
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| Vydané v: | Foundations of computational mathematics Ročník 19; číslo 3; s. 703 - 773 |
|---|---|
| Hlavní autori: | , |
| Médium: | Journal Article |
| Jazyk: | English |
| Vydavateľské údaje: |
New York
Springer US
01.06.2019
Springer Nature B.V |
| Predmet: | |
| ISSN: | 1615-3375, 1615-3383 |
| On-line prístup: | Získať plný text |
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| Shrnutí: | In phase retrieval, we want to recover an unknown signal
x
∈
C
d
from
n
quadratic measurements of the form
y
i
=
|
⟨
a
i
,
x
⟩
|
2
+
w
i
, where
a
i
∈
C
d
are known sensing vectors and
w
i
is measurement noise. We ask the following
weak recovery
question: What is the minimum number of measurements
n
needed to produce an estimator
x
^
(
y
)
that is positively correlated with the signal
x
? We consider the case of Gaussian vectors
a
i
. 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
n
≤
d
-
o
(
d
)
, no estimator can do significantly better than random and achieve a strictly positive correlation. For
n
≥
d
+
o
(
d
)
, 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
y
i
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. |
|---|---|
| Bibliografia: | ObjectType-Article-1 SourceType-Scholarly Journals-1 ObjectType-Feature-2 content type line 14 |
| ISSN: | 1615-3375 1615-3383 |
| DOI: | 10.1007/s10208-018-9395-y |