THESIS
2021

1 online resource (xii, 105 pages) : illustrations (some color)

**Abstract**
This thesis is to study theories and efficient algorithms for sparse phase retrieval
problem, and two non-convex methods are introduced in the work. The sparse
phase retrieval problem is to recover an s-sparse signal x

^{♮} ∈ R

^{n } (s n) from m
phaseless samples y

_{i} = │❬x

^{♮},a

_{i}❭│ for i = 1, . . . ,m. Sparse phase retrieval problem
arises in many applications related to signal and image processing. The task of
the work is to develop efficient algorithms and theories for the problem using
few number of measurements, especially in case of m n. Existing gradient
descent type methods for sparse phase retrieval are first-order and hence converge
at most linearly. In the thesis, firstly, a Newton's approach named HTP
for sparse phase retrieval has been introduced, which is guaranteed to find the
und...[

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This thesis is to study theories and efficient algorithms for sparse phase retrieval
problem, and two non-convex methods are introduced in the work. The sparse
phase retrieval problem is to recover an s-sparse signal x

^{♮} ∈ R

^{n } (s << n) from m
phaseless samples y

_{i} = │❬x

^{♮},a

_{i}❭│ for i = 1, . . . ,m. Sparse phase retrieval problem
arises in many applications related to signal and image processing. The task of
the work is to develop efficient algorithms and theories for the problem using
few number of measurements, especially in case of m < n. Existing gradient
descent type methods for sparse phase retrieval are first-order and hence converge
at most linearly. In the thesis, firstly, a Newton's approach named HTP
for sparse phase retrieval has been introduced, which is guaranteed to find the
underlying signal in finite steps using only m ～ O(s log(n/s)) Gaussian samples
and the initialization is in a neighbourhood of the underlying sparse signal.
Together with a spectral initialization, HTP is guaranteed to have an exact recovery
from O(s

^{2} log n) samples. While the computational cost per iteration is
the same order as first order methods, the proposed algorithm is quite efficient
in terms of cpu time and numerical experiments illustrates that the proposed
HTP is several times faster than state-of-the-art algorithms. Further, based on a
standard alternating minimization for sparse phase retrieval, a novel stochastic version of alternating minimization method for the problem has been proposed
in the work. By introducing a generalized RIP condition on the sensing matrix,
theoretical justification for the finite step convergence (in at most O(log m)
steps) of the proposed stochastic algorithm is provided. Numerical experiments
show that the proposed stochastic algorithm requires less measurements than
state-of-the-art algorithms.

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