24 research outputs found
Sparse Power Factorization: Balancing peakiness and sample complexity
In many applications, one is faced with an inverse problem, where the known
signal depends in a bilinear way on two unknown input vectors. Often at least
one of the input vectors is assumed to be sparse, i.e., to have only few
non-zero entries. Sparse Power Factorization (SPF), proposed by Lee, Wu, and
Bresler, aims to tackle this problem. They have established recovery guarantees
for a somewhat restrictive class of signals under the assumption that the
measurements are random. We generalize these recovery guarantees to a
significantly enlarged and more realistic signal class at the expense of a
moderately increased number of measurements.Comment: 18 page
Auto-Calibration and Biconvex Compressive Sensing with Applications to Parallel MRI
We study an auto-calibration problem in which a transform-sparse signal is
compressive-sensed by multiple sensors in parallel with unknown sensing
parameters. The problem has an important application in pMRI reconstruction,
where explicit coil calibrations are often difficult and costly to achieve in
practice, but nevertheless a fundamental requirement for high-precision
reconstructions. Most auto-calibrated strategies result in reconstruction that
corresponds to solving a challenging biconvex optimization problem. We
transform the auto-calibrated parallel sensing as a convex optimization problem
using the idea of `lifting'. By exploiting sparsity structures in the signal
and the redundancy introduced by multiple sensors, we solve a mixed-norm
minimization problem to recover the underlying signal and the sensing
parameters simultaneously. Robust and stable recovery guarantees are derived in
the presence of noise and sparsity deficiencies in the signals. For the pMRI
application, our method provides a theoretically guaranteed approach to
self-calibrated parallel imaging to accelerate MRI acquisitions under
appropriate assumptions. Developments in MRI are discussed, and numerical
simulations using the analytical phantom and simulated coil sensitives are
presented to support our theoretical results.Comment: Keywords: Self-calibration, Compressive sensing, Convex optimization,
Random matrices, Parallel MR
Regularized Gradient Descent: A Nonconvex Recipe for Fast Joint Blind Deconvolution and Demixing
We study the question of extracting a sequence of functions
from observing only the sum of
their convolutions, i.e., from . While convex optimization techniques
are able to solve this joint blind deconvolution-demixing problem provably and
robustly under certain conditions, for medium-size or large-size problems we
need computationally faster methods without sacrificing the benefits of
mathematical rigor that come with convex methods. In this paper, we present a
non-convex algorithm which guarantees exact recovery under conditions that are
competitive with convex optimization methods, with the additional advantage of
being computationally much more efficient. Our two-step algorithm converges to
the global minimum linearly and is also robust in the presence of additive
noise. While the derived performance bounds are suboptimal in terms of the
information-theoretic limit, numerical simulations show remarkable performance
even if the number of measurements is close to the number of degrees of
freedom. We discuss an application of the proposed framework in wireless
communications in connection with the Internet-of-Things.Comment: Accepted to Information and Inference: a Journal of the IM