| Summary: | We consider the problem of reconstructing an
n-dimensional k-sparse signal from a set of
noiseless magnitude-only measurements. Formulating the problem as an unregularized empirical risk minimization task, we study the
sample complexity performance of gradient descent with Hadamard parametrization, which
we call Hadamard Wirtinger flow (HWF). Provided knowledge of the signal sparsity k, we
prove that a single step of HWF is able to
recover the support from k(x
∗
max)
−2
(modulo
logarithmic term) samples, where x
∗
max is the
largest component of the signal in magnitude.
This support recovery procedure can be used
to initialize existing reconstruction methods
and yields algorithms with total runtime proportional to the cost of reading the data and
improved sample complexity, which is linear
in k when the signal contains at least one large
component. We numerically investigate the
performance of HWF at convergence and show
that, while not requiring any explicit form
of regularization nor knowledge of k, HWF
adapts to the signal sparsity and reconstructs
sparse signals with fewer measurements than
existing gradient based methods.
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