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 $l_1$-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