Compressive Sensing via Variational Bayesian Inference under Two Widely Used Priors: Modeling, Comparison and Discussion.

Compressive sensing is a sub-Nyquist sampling technique for efficient signal acquisition and reconstruction of sparse or compressible signals. In order to account for the sparsity of the underlying signal of interest, it is common to use sparsifying priors such as Bernoulli-Gaussian-inverse Gamma (BGiG) and Gaussian-inverse Gamma (GiG) priors on the components of the signal. With the introduction of variational Bayesian inference, the sparse Bayesian learning (SBL) methods for solving the inverse problem of compressive sensing have received significant interest as the SBL methods become more efficient in terms of execution time. In this paper, we consider the sparse signal recovery problem using compressive sensing and the variational Bayesian (VB) inference framework. More specifically, we consider two widely used Bayesian models of BGiG and GiG for modeling the underlying sparse signal for this problem. Although these two models have been widely used for sparse recovery problems under various signal structures, the question of which model can outperform the other for sparse signal recovery under no specific structure has yet to be fully addressed under the VB inference setting. Here, we study these two models specifically under VB inference in detail, provide some motivating examples regarding the issues in signal reconstruction that may occur under each model, perform comparisons and provide suggestions on how to improve the performance of each model.

Bayesian Compressive Sensing of Sparse Signals with Unknown Clustering Patterns.

We consider the sparse recovery problem of signals with an unknown clustering pattern in the context of multiple measurement vectors (MMVs) using the compressive sensing (CS) technique. For many MMVs in practice, the solution matrix exhibits some sort of clustered sparsity pattern, or clumpy behavior, along each column, as well as joint sparsity across the columns. In this paper, we propose a new sparse Bayesian learning (SBL) method that incorporates a total variation-like prior as a measure of the overall clustering pattern in the solution. We further incorporate a parameter in this prior to account for the emphasis on the amount of clumpiness in the supports of the solution to improve the recovery performance of sparse signals with an unknown clustering pattern. This parameter does not exist in the other existing algorithms and is learned via our hierarchical SBL algorithm. While the proposed algorithm is constructed for the MMVs, it can also be applied to the single measurement vector (SMV) problems. Simulation results show the effectiveness of our algorithm compared to other algorithms for both SMV and MMVs.

Exploration vs. Data Refinement via Multiple Mobile Sensors.

We examine the deployment of multiple mobile sensors to explore an unknown region to map regions containing concentration of a physical quantity such as heat, electron density, and so on. The exploration trades off between two desiderata: to continue taking data in a region known to contain the quantity of interest with the intent of refining the measurements vs. taking data in unobserved areas to attempt to discover new regions where the quantity may exist. Making reasonable and practical decisions to simultaneously fulfill both goals of exploration and data refinement seem to be hard and contradictory. For this purpose, we propose a general framework that makes value-laden decisions for the trajectory of mobile sensors. The framework employs a Gaussian process regression model to predict the distribution of the physical quantity of interest at unseen locations. Then, the decision-making on the trajectories of sensors is performed using an epistemic utility controller. An example is provided to illustrate the merit and applicability of the proposed framework.

EXPLORATION AND DATA REFINEMENT VIA MULTIPLE MOBILE SENSORS BASED ON GAUSSIAN PROCESSES.

We consider configuration of multiple mobile sensors to explore and refine knowledge in an unknown field. After some initial discovery, it is desired to collect data from the regions that are far away from the current sensor trajectories in order to favor the exploration purposes, while simultaneously, exploring the vicinity of known interesting phenomena to refine the measurements. Since the collected data only provide us with local information, there is no optimal solution to be sought for the next trajectory of sensors. Using Gaussian process regression, we provide a simple framework that accounts for both the conflicting data refinement and exploration goals, and to make reasonable decisions for the trajectories of mobile sensors.

AMP-B-SBL: An algorithm for clustered sparse signals using approximate message passing.

Recently, we proposed an algorithm for the single measurement vector problem where the underlying sparse signal has an unknown clustered pattern. The algorithm is essentially a sparse Bayesian learning (SBL) algorithm simplified via the approximate message passing (AMP) framework. Treating the cluster pattern is controlled via a knob that accounts for the amount of clumpiness in the solution. The parameter corresponding to the knob is learned using expectation-maximization algorithm. In this paper, we provide further study by comparing the performance of our algorithm with other algorithms in terms of support recovery, mean-squared error, and an example in image reconstruction in a compressed sensing fashion.

SPARSE BAYESIAN LEARNING BOOSTED BY PARTIAL ERRONEOUS SUPPORT KNOWLEDGE.

Recovery of sparse signals with unknown clustering pattern in the case of having partial erroneous prior knowledge on the supports of the signal is considered. In this case, we provide a modified sparse Bayesian learning model to incorporate prior knowledge and simultaneously learn the unknown clustering pattern. For this purpose, we add one more layer to support-aided sparse Bayesian learning algorithm (SA-SBL). This layer adds a prior on the shape parameters of Gamma distributions, those modeled to account for the precision of the solution elements. We make the shape parameters depend on the total variations on the estimated supports of the solution. Based on the simulation results, we show that the proposed algorithm is able to modify its erroneous prior knowledge on the supports of the solution and learn the clustering pattern of the true signal by filtering out the incorrect supports from the estimated support set.

On The Block-Sparse Solution of Single Measurement Vectors.

Finding the solution of single measurement vector (SMV) problem with an unknown block-sparsity structure is considered. Here, we propose a sparse Bayesian learning (SBL) algorithm simplified via the approximate message passing (AMP) framework. In order to encourage the block-sparsity structure, we incorporate a parameter called Sigma-Delta as a measure of clumpiness in the supports of the solution. Using the AMP framework reduces the computational load of the proposed SBL algorithm and as a result makes it faster. Furthermore, in terms of the mean-squared error between the true and the reconstructed solution, the algorithm demonstrates an encouraging improvement compared to the other algorithms.

ON THE BLOCK-SPARSITY OF MULTIPLE-MEASUREMENT VECTORS.

Based on the compressive sensing (CS) theory, it is possible to recover signals, which are either compressible or sparse under some suitable basis, via a small number of non-adaptive linear measurements. In this paper, we investigate recovering of block-sparse signals via multiple measurement vectors (MMVs) in the presence of noise. In this case, we consider one of the existing algorithms which provides a satisfactory estimate in terms of minimum mean-squared error but a non-sparse solution. Here, the algorithm is first modified to result in sparse solutions. Then, further modification is performed to account for the unknown block sparsity structure in the solution, as well. The performance of the proposed algorithm is demonstrated by experimental simulations and comparisons with some other algorithms for the sparse recovery problem.

Hierarchical Bayesian Approach For Jointly-Sparse Solution Of Multiple-Measurement Vectors.

It is well-known that many signals of interest can be well-estimated via just a small number of supports under some specific basis. Here, we consider finding sparse solution for Multiple Measurement Vectors (MMVs) in case of having both jointly sparse and clumpy structure. Most of the previous work for finding such sparse representations are based on greedy and sub-optimal algorithms such as Basis Pursuit (BP), Matching Pursuit (MP), and Orthogonal Matching Pursuit (OMP). In this paper, we first propose a hierarchical Bayesian model to deal with MMVs that have jointly-sparse structure in their solutions. Then, the model is modified to account for clumps of the neighbor supports (block sparsity) in the solution structure, as well. Several examples are considered to illustrate the merit of the proposed hierarchical Bayesian model compared to OMP and a modified version of the OMP algorithm.