4 research outputs found
A New Analysis of Compressive Sensing by Stochastic Proximal Gradient Descent
In this manuscript, we analyze the sparse signal recovery (compressive
sensing) problem from the perspective of convex optimization by stochastic
proximal gradient descent. This view allows us to significantly simplify the
recovery analysis of compressive sensing. More importantly, it leads to an
efficient optimization algorithm for solving the regularized optimization
problem related to the sparse recovery problem. Compared to the existing
approaches, there are two advantages of the proposed algorithm. First, it
enjoys a geometric convergence rate and therefore is computationally efficient.
Second, it guarantees that the support set of any intermediate solution
generated by the proposed algorithm is concentrated on the support set of the
optimal solution
Bayesian Compressive Sensing with Circulant Matrix for Spectrum Sensing in Cognitive Radio Networks
For wideband spectrum sensing, compressive sensing has been proposed as a
solution to speed up the high dimensional signals sensing and reduce the
computational complexity. Compressive sensing consists of acquiring the
essential information from a sparse signal and recovering it at the receiver
based on an efficient sampling matrix and a reconstruction technique. In order
to deal with the uncertainty, improve the signal acquisition performance, and
reduce the randomness during the sensing and reconstruction processes,
compressive sensing requires a robust sampling matrix and an efficient
reconstruction technique. In this paper, we propose an approach that combines
the advantages of a Circulant matrix with Bayesian models. This approach is
implemented, extensively tested, and its results have been compared to those of
l1 norm minimization with a Circulant or random matrix based on several
metrics. These metrics are Mean Square Error, reconstruction error,
correlation, recovery time, sampling time, and processing time. The results
show that our technique is faster and more efficient
Decomposable Norm Minimization with Proximal-Gradient Homotopy Algorithm
We study the convergence rate of the proximal-gradient homotopy algorithm
applied to norm-regularized linear least squares problems, for a general class
of norms. The homotopy algorithm reduces the regularization parameter in a
series of steps, and uses a proximal-gradient algorithm to solve the problem at
each step. Proximal-gradient algorithm has a linear rate of convergence given
that the objective function is strongly convex, and the gradient of the smooth
component of the objective function is Lipschitz continuous. In many
applications, the objective function in this type of problem is not strongly
convex, especially when the problem is high-dimensional and regularizers are
chosen that induce sparsity or low-dimensionality. We show that if the linear
sampling matrix satisfies certain assumptions and the regularizing norm is
decomposable, proximal-gradient homotopy algorithm converges with a
\emph{linear rate} even though the objective function is not strongly convex.
Our result generalizes results on the linear convergence of homotopy algorithm
for -regularized least squares problems. Numerical experiments are
presented that support the theoretical convergence rate analysis
Compressive Spectrum Sensing for Cognitive Radio Networks
A cognitive radio system has the ability to observe and learn from the
environment, adapt to the environmental conditions, and use the radio spectrum
more efficiently. It allows secondary users (SUs) to use the primary users
(PUs) channels when they are not being utilized. Cognitive radio involves three
main processes: spectrum sensing, deciding, and acting. In the spectrum sensing
process, the channel occupancy is measured with spectrum sensing techniques in
order to detect unused channels. In the deciding process, sensing results are
analyzed and decisions are made based on these results. In the acting process,
actions are made by adjusting the transmission parameters to enhance the
cognitive radio performance.
One of the main challenges of cognitive radio is the wideband spectrum
sensing. Existing spectrum sensing techniques are based on a set of
observations sampled by an ADC at the Nyquist rate. However, those techniques
can sense only one channel at a time because of the hardware limitations on the
sampling rate. In addition, in order to sense a wideband spectrum, the wideband
is divided into narrow bands or multiple frequency bands. SUs have to sense
each band using multiple RF frontends simultaneously, which can result in a
very high processing time, hardware cost, and computational complexity. In
order to overcome this problem, the signal sampling should be as fast as
possible even with high dimensional signals. Compressive sensing has been
proposed as a low-cost solution to reduce the processing time and accelerate
the scanning process. It allows reducing the number of samples required for
high dimensional signal acquisition while keeping the essential information.Comment: PhD dissertation, Advisors: Dr. Naima Kaabouch and Dr. Hassan El
Ghaz