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

Robust Bayesian compressed sensing with outliers

Variational Bayesian inference based robust multiple measurement sparse signal recovery

Sparse Bayesian learning using correlated hyperparameters for recovery of block sparse signals

In this article we study the sparse signal recovery problem in a Bayesian framework using a novel Bootstrapped Sparse Bayesian Learning method. Sparse Bayesian Learning (SBL) framework is an effective tool for pruning out the irrelevant features and ending up with a sparse representation. In SBL the choice of prior over the variances of the Gaussian Scale mixture has been an interesting area of research for some time now. This motivates us to use a more generalized maximum entropy density as the prior which results in a new variant of SBL. It has been shown to perform better than traditional SBL empirically and it also accelerates the pruning procedure. Because of this advantage, this variant of SBL can be claimed as more robust choice as it is less sensitive to the threshold for pruning. Theoretical justifications have also been provided to show that the proposed model actually promotes sparse point estimates.

We investigate the performance of sparse signal recovery from a set of compressively sensed noisy measurements using variational Bayesian (VB) inference. The framework considered here is sparse Bayesian learning with no specific structure other than sparsity assumed on the signal. Two models are considered here, the issues of each model is studied, and the performances in reconstruction are compared. These models have been already applied to the compressive sensing problems implemented via expectation-maximization algorithm, Markov chain Monte Carlo (MCMC), etc., for different structures of the solution. However, the issues when using variational Bayes inference for only promoting the sparsity have not been investigated. This paper provides details on such modeling and the issues accompanied with each modeling are discussed.

This letter addresses the recovery of hierarchical sparse signals in a Bayesian framework. Hierarchical sparse signals exhibit two levels of sparsity, i.e., block-sparsity among different blocks and internal sparsity within each individual block. As in sparse Bayesian learning, each component of the coefficient vector is firstly modeled as a Gaussian distributed variable with zero mean. To enforce the two-level hierarchical sparsity, the variance is further modeled by two classes of hidden variables controlling the block-sparsity and the internal sparsity, respectively. Finally, variational Bayesian inference is used to recover the coefficient vector from the noise corrupted data. Numerical simulation and experimental results show that the proposed method outperforms those recently reported recovery methods.

Aiming at DOA estimation under impulsive noise, this paper propose a real-valued sparse Bayesian learning (SBL) method. A unitary transformation is utilized to convert complex-valued direction-of-arrival (DOA) estimation into real ones. The variational Bayesian inference (VBI) technique is then adopted to perform the Bayesian inference with such real-valued prior. Consequently, the computational complexity of this Bayesian inference is significantly reduced. Simulation outcomes demonstrate the great robust performance and low computational load of the new method.

The conditions under which Compressed sensing (CS) succeeds depend on the structure of the measurement matrix. Researches indicated that matrices whose entries are drawn independently from certain probability distributions satisfy the restricted isometry property (RIP) and guarantee exact recovery of a sparse signal with high probability. Motivated by signal processing applications, random filtering with Toeplitz sensing matrices whose elements are drawn from the same distributions were considered and shown to also be sufficient to recover a sparse signal from reduced samples exactly with high probability. In order to achieve reduction of sampling rate in impulse radio Ultra-Wideband (IR-UWB) communication systems, the output streams of UWB channel were periodically down-sampled. Sub-sampling at the receiver leads to quasi-Toeplitz-structured measurement matrices, whose entries are UWB channel coefficients. In this paper, the feasibility of this kind of quasi-Toeplitz matrices as measurement matrices is discussed. It is shown that the quasi-Toeplitz matrices with entries drawn from logarithm normal distribution satisfy RIP with high probability and also ensures the exact reconstruction of the sparse signals. Simulation results validate their performance.

Orthogonal time frequency space (OTFS) is a new emerging modulation scheme that performs better than orthogonal frequency division multiplexing (OFDM) in high mobility scenarios. In this paper, we consider the delay-Doppler (DD) channel estimation problem in an OTFS system. By exploiting the inherent sparse nature of the DD channel, the channel estimation problem is modeled as a sparse signal recovery problem. Next, we build a two-layer graphical model with the Laplacian scale mixture (LSM) prior utilized to model the sparse channel. Then, a variational Bayesian inference (VBI) based algorithm is proposed to solve this problem. Simulation results are presented to show that the proposed algorithm can achieve better performance than other existing channel estimation algorithms.

This study concerns the issue of jointly enhancing noise robustness and promoting signal sparsity in Sparse Bayesian Learning (SBL), which aims at addressing the performance deficiency of sparse signal recovery due to uninformative data with low signal-to-noise ratios. In particular, the authors propose a hierarchical prior noise model with a signal-dependent parametrisation and incorporate it into developing the robust SBL algorithms for sparse signal recovery. The main contribution of the proposed approach is twofold. The first is the new consideration of noise-robustness enhancement in building SBL algorithms, which devotes to noise awareness in counteracting outliers in measurements. Specifically, the idea of signal-sparsity enforcing is extended to build a Least Absolute Deviation like loss criterion with the proposed hierarchical prior model of measurement noise. The second is the novelty of using the signal-dependent parametrisation in the proposed noise model. Indeed, the signal-dependent mechanism plays an indispensable role in producing the reliable noise parameter estimation jointly with updating signal model parameters under the fast SBL framework. In addition to numerical simulation studies, the real-life application of radio tomographic imaging is presented to validate the proposed approach.

Global sensitivity analysis: A Bayesian learning based polynomial chaos approach

Cognitive Radio (CR) is one of the most promising techniques for optimizing the spectrum usage. However, the large amount of data of spectral information that must be processed to identify and assign spectral resources increases the channel assignment times, therefore worsening the quality of service for the devices using the spectrum. Compressive Sensing (CS) is a digital processing technique that allows the reconstruction of sparse or compressible signals using fewer samples than those required traditionally. This paper presents a model that addresses the Spectral Sensing problem in Cognitive Radio using Compressive Sensing as an effective way of decreasing the number of samples required in the sensing process. This model is based on Compressive Spectral Imaging (CSI) architectures where a centralized spectrum manager selects what power data must be delivered by the different wireless devices using binary patterns, and builds a multispectral data cube image with the geographical and spectral data power information. The results show that this multispectral data cube can be built with only a 50% of the samples generated by the devices and, therefore reducing the data traffic dramatically.

Fast acquisition in magnetic resonance imaging (MRI) is considered in this paper. Often, fast acquisition is achieved using parallel imaging (pMRI) techniques. It has been shown recently that the combination of pMRI and compressed sensing (CS), which enables exact reconstruction of sparse or compressible signals from a small number of random measurements, can accelerate the speed of MRI acquisition because the number of measurements are much smaller than that by pMRI per se. Also recently in CS, chaos filters were designed to obtain chaotic measurements. This chaotic CS approach potentially offers simpler hardware implementation. In this paper, we combine chaotic CS and pMRI. However, instead of using chaos filters, the measurements are obtained by chaotically undersampling the k-space. MRI image reconstruction is then performed by using nonlinear conjugate gradient optimization. For pMRI, we use the well-known approach SENSE - sensitivity encoding -, which requires an estimation of the sensitivity maps. The performance of the proposed method is analyzed using the point spread function, the transform point spread function, and the reconstruction error measure.

Compressive sensing (CS) idea enables the reconstruction of a sparse signal from a small set of measurements. CS approach has applications in many practical areas. One of the areas is radar systems. In this article, the radar ambiguity function is denoised within the CS framework. A new denoising method on the projection onto the epigraph set of the convex function is also developed for this purpose. This approach is compared to the other CS reconstruction algorithms. Experimental results are presented1.

Sparse signal priors help in a variety of modern signal processing tasks. In many cases, a sparse signal needs to be recovered from an underdetermined system of equations. For instance, sparse approximation of a signal with an overcomplete dictionary or reconstruction of a sparse signal from a small number of linear measurements. The reconstruction problem typically requires solving an ℓ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> norm minimization problem. In this paper we present homotopy based algorithms to update the solution of some ℓ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> problems when the system is updated by adding new rows or columns to the underlying system matrix. We also discuss a case where these ideas can be extended to accommodate for more general changes in the system matrix.