In this paper, we consider a diffusion network with the objective of minimizing a cost function at each node by a global optimizer, assumed to be K-sparse, by utilizing diffusion based collaboration among the nodes. In the proposed algorithm termed as DiffstoHT here, in each iteration, first a stochastic gradient based update is calculated followed by K level hard thresholding at each node. The resulting estimates are then shared with the neighbors of the node by means of diffusion and subsequently a weighted linear combination of the incoming estimates is taken followed by 2K level hard thresholding. Next, each node carries out certain gradient descent iterations sequentially, restricted to the 2K-sized support obtained from previous stage. Lastly, another K level hard thresholding is performed by each node, completing one iteration. The proposed DiffStoHT algorithm is particularly useful for large scale problems where full gradient calculation is expensive. Theoretical analysis of the proposed DiffStoHT algorithm is carried out using the restricted positive definite Hessian (RPDH) property of the cost functions and sufficient conditions for recovery are derived. Simulations suggest that the proposed algorithm is not only faster than existing diffusion based methods, but it is also capable of outperforming them in terms of probability of recovery.

This paper presents a unified treatment to hard thresholding based compressed sensing recovery algorithms. For this, it modifies the cost function by a power series in$\mathbf {A}\mathbf {A}^{H}$where$\mathbf {A}$is the so-called sensing matrix. For appropriate choice of the power series coefficients, the proposed treatment not only leads to various existing hard thresholding based recovery algorithms, but, more importantly, it enables one to develop new algorithms belonging to this category. The paper also presents a convergence analysis of the proposed method and derives convergence guarantees in terms of the restricted isometry constant (RIC) of the sensing matrix. It is seen that in case of some of the well known hard thresholding based algorithms, the proposed convergence analysis results in wider ranges of algorithm parameters and faster convergence than suggested by existing analyses. Some new, power series based hard thresholding algorithms are also proposed and their recovery performance studied via simulation.

In this paper, we propose a Hard Thresholding (HT) based robust algorithm for large scale Multiple Measurement Vectors (MMV) problem in Compressed Sensing. The proposed MStoLIHT algorithm is based upon Stochastic Gradient Descent (SGD) and employs the Lorentzian norm of residual as the underlying cost function which provides robustness against impulsive errors in the measurements. Each iteration of the MStoLIHT algorithm consists of multiple subiterations. At each iteration, only a block of rows of the sensing matrix is used for carrying out the update, and at each subiteration, only few of the columns of the estimated data matrix are updated based on the SGD and the HT strategies. At the end of subiterations, another HT operation is performed ensuring the row sparsity in the estimated data matrix. Extensive numerical simulations are carried out, which indicate that by updating the data matrix in blocks of columns followed by HT operation, the performance in terms of probability of support recovery is improved when compared to its deterministic counterpart. It is further observed that the MStoLIHT algorithm outperforms other existing algorithms for the MMV problem in literature.

In this paper, we first present a stochastic gradient based robust algorithm for recovering a sparse signal from compressed measurements corrupted by impulsive noise for large problems where calculation of the full gradient is expensive. This stochastic gradient based strategy is then modified and applied to diffusion based distributed compressed sensing. In the proposed algorithm, a proxy to the actual gradient is found and hard thresholding based updates are carried out. The proposed algorithm uses Lorentzian norm of the residual as the cost function, making it robust against impulsive noise. It is observed through simulations that the proposed algorithm is able to outperform existing stochastic gradient based algorithms and is able to provide at par recovery performance to that of other robust deterministic algorithms currently available in literature for distributed compressed sensing.

In this paper, we present two heavy ball based hard thresholding algorithms aimed at recovering jointly sparse signals in multiple measurement vector (MMV) scenario, arising in compressed sensing. The proposed Simultaneous Heavy Ball based Iterative Hard Thresholding (SHBIHT) and Simultaneous Heavy Ball based Hard Thresholding Pursuit (SHBHTP) algorithms use heavy ball based acceleration technique, which uses the current estimate as well as the previous estimate in the gradient based update. By exploiting the MMV structure, we use a weighted momentum rather than a common momentum for each of the signals. In the first algorithm, hard thresholding is carried out on the gradient based update whereas the other algorithm requires solving a least squares problem (pursuit step) on top of the hard thresholded update. Theoretical analysis is carried out using the Restricted Isometry Property (RIP) and sufficient conditions for convergence are derived. It is observed through simulations that the proposed heavy ball based algorithms for MMV problem provide computational advantage in terms of total time required for convergence while performing at-par with existing algorithms in terms of recovery performance.

In this paper, we present two robust, thresholding based distributed algorithms for recovering a sparse signal from compressed measurements. These algorithms use Lorentzian norm as the cost function which has been observed to give robustness against heavy tailed noise, e.g., impulsive noise. The first algorithm named Diffusion based Lorentzian Iterative Hard Thresholding (DLIHT) requires only gradient information whereas the second algorithm named Diffusion based Lorentzian Hard Thresholding Pursuit(DLHTP) requires solving a linear system, on top of the gradient based update. It is observed through simulations that for moderate corruptions, DLHTP converges faster than DLIHT to similar steady state error value. However, for higher levels of corruptions, DLIHT leads to much less steady state error than offered by DLHTP.