Abstract
Abstract
The phase retrieval problem is concerned with recovering an unknown signal
x
∈
C
n
from a set of magnitude-only measurements
y
j
=
∣
⟨
a
j
,
x
⟩
∣
,
j
=
1
,
…
,
m
. A natural least squares formulation can be used to solve this problem efficiently even with random initialization, despite its non-convexity of the loss function. One way to explain this surprising phenomenon is the benign geometric landscape: (1) all local minimizers are global; and (2) the objective function has a negative curvature around each saddle point and local maximizer. In this paper, we show that
m
=
O
(
n
log
n
)
Gaussian random measurements are sufficient to guarantee the loss function of a commonly used estimator has such benign geometric landscape with high probability. This is a step toward answering the open problem given by Sun et al (2018 Found. Comput. Math.
18 1131–98), in which the authors suggest that
O
(
n
log
n
)
or even O(n) is enough to guarantee the favorable geometric property.
Funder
Hong Kong Research Grant Council grants
Subject
Applied Mathematics,Computer Science Applications,Mathematical Physics,Signal Processing,Theoretical Computer Science