146 research outputs found

    Cramer Rao-Type Bounds for Sparse Bayesian Learning

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    In this paper, we derive Hybrid, Bayesian and Marginalized Cram\'{e}r-Rao lower bounds (HCRB, BCRB and MCRB) for the single and multiple measurement vector Sparse Bayesian Learning (SBL) problem of estimating compressible vectors and their prior distribution parameters. We assume the unknown vector to be drawn from a compressible Student-t prior distribution. We derive CRBs that encompass the deterministic or random nature of the unknown parameters of the prior distribution and the regression noise variance. We extend the MCRB to the case where the compressible vector is distributed according to a general compressible prior distribution, of which the generalized Pareto distribution is a special case. We use the derived bounds to uncover the relationship between the compressibility and Mean Square Error (MSE) in the estimates. Further, we illustrate the tightness and utility of the bounds through simulations, by comparing them with the MSE performance of two popular SBL-based estimators. It is found that the MCRB is generally the tightest among the bounds derived and that the MSE performance of the Expectation-Maximization (EM) algorithm coincides with the MCRB for the compressible vector. Through simulations, we demonstrate the dependence of the MSE performance of SBL based estimators on the compressibility of the vector for several values of the number of observations and at different signal powers.Comment: Accepted for publication in the IEEE Transactions on Signal Processing, 11 pages, 10 figure

    Fast Variational Block-Sparse Bayesian Learning

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    We present a fast update rule for variational block-sparse Bayesian learning (SBL) methods. Using a variational Bayesian framework, we show how repeated updates of probability density functions (PDFs) of the prior variances and weights can be expressed as a nonlinear first-order recurrence from one estimate of the parameters of the proxy PDFs to the next. Specifically, the recurrent relation turns out to be a strictly increasing rational function for many commonly used prior PDFs of the variances, such as Jeffrey's prior. Hence, the fixed points of this recurrent relation can be obtained by solving for the roots of a polynomial. This scheme allows to check for convergence/divergence of individual prior variances in a single step. Thereby, the the computational complexity of the variational block-SBL algorithm is reduced and the convergence speed is improved by two orders of magnitude in our simulations. Furthermore, the solution allows insights into the sparsity of the estimators obtained by choosing different priors.Comment: 10 pages, 2 figures, submitted to IEEE Transactions on Signal Processing on 1st of June, 202

    Sparse Signal Recovery Based on Compressive Sensing and Exploration Using Multiple Mobile Sensors

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    The work in this dissertation is focused on two areas within the general discipline of statistical signal processing. First, several new algorithms are developed and exhaustively tested for solving the inverse problem of compressive sensing (CS). CS is a recently developed sub-sampling technique for signal acquisition and reconstruction which is more efficient than the traditional Nyquist sampling method. It provides the possibility of compressed data acquisition approaches to directly acquire just the important information of the signal of interest. Many natural signals are sparse or compressible in some domain such as pixel domain of images, time, frequency and so forth. The notion of compressibility or sparsity here means that many coefficients of the signal of interest are either zero or of low amplitude, in some domain, whereas some are dominating coefficients. Therefore, we may not need to take many direct or indirect samples from the signal or phenomenon to be able to capture the important information of the signal. As a simple example, one can think of a system of linear equations with N unknowns. Traditional methods suggest solving N linearly independent equations to solve for the unknowns. However, if many of the variables are known to be zero or of low amplitude, then intuitively speaking, there will be no need to have N equations. Unfortunately, in many real-world problems, the number of non-zero (effective) variables are unknown. In these cases, CS is capable of solving for the unknowns in an efficient way. In other words, it enables us to collect the important information of the sparse signal with low number of measurements. Then, considering the fact that the signal is sparse, extracting the important information of the signal is the challenge that needs to be addressed. Since most of the existing recovery algorithms in this area need some prior knowledge or parameter tuning, their application to real-world problems to achieve a good performance is difficult. In this dissertation, several new CS algorithms are proposed for the recovery of sparse signals. The proposed algorithms mostly do not require any prior knowledge on the signal or its structure. In fact, these algorithms can learn the underlying structure of the signal based on the collected measurements and successfully reconstruct the signal, with high probability. The other merit of the proposed algorithms is that they are generally flexible in incorporating any prior knowledge on the noise, sparisty level, and so on. The second part of this study is devoted to deployment of mobile sensors in circumstances that the number of sensors to sample the entire region is inadequate. Therefore, where to deploy the sensors, to both explore new regions while refining knowledge in aleady visited areas is of high importance. Here, a new framework is proposed to decide on the trajectories of sensors as they collect the measurements. The proposed framework has two main stages. The first stage performs interpolation/extrapolation to estimate the phenomenon of interest at unseen loactions, and the second stage decides on the informative trajectory based on the collected and estimated data. This framework can be applied to various problems such as tuning the constellation of sensor-bearing satellites, robotics, or any type of adaptive sensor placement/configuration problem. Depending on the problem, some modifications on the constraints in the framework may be needed. As an application side of this work, the proposed framework is applied to a surrogate problem related to the constellation adjustment of sensor-bearing satellites

    Compression via Compressive Sensing : A Low-Power Framework for the Telemonitoring of Multi-Channel Physiological Signals

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    Telehealth and wearable equipment can deliver personal healthcare and necessary treatment remotely. One major challenge is transmitting large amount of biosignals through wireless networks. The limited battery life calls for low-power data compressors. Compressive Sensing (CS) has proved to be a low-power compressor. In this study, we apply CS on the compression of multichannel biosignals. We firstly develop an efficient CS algorithm from the Block Sparse Bayesian Learning (BSBL) framework. It is based on a combination of the block sparse model and multiple measurement vector model. Experiments on real-life Fetal ECGs showed that the proposed algorithm has high fidelity and efficiency. Implemented in hardware, the proposed algorithm was compared to a Discrete Wavelet Transform (DWT) based algorithm, verifying the proposed one has low power consumption and occupies less computational resources.Comment: 2013 International Workshop on Biomedical and Health Informatic

    Bayesian Compressive Sensing of Sparse Signals with Unknown Clustering Patterns

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    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

    Variational Bayesian Inference of Line Spectra

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    In this paper, we address the fundamental problem of line spectral estimation in a Bayesian framework. We target model order and parameter estimation via variational inference in a probabilistic model in which the frequencies are continuous-valued, i.e., not restricted to a grid; and the coefficients are governed by a Bernoulli-Gaussian prior model turning model order selection into binary sequence detection. Unlike earlier works which retain only point estimates of the frequencies, we undertake a more complete Bayesian treatment by estimating the posterior probability density functions (pdfs) of the frequencies and computing expectations over them. Thus, we additionally capture and operate with the uncertainty of the frequency estimates. Aiming to maximize the model evidence, variational optimization provides analytic approximations of the posterior pdfs and also gives estimates of the additional parameters. We propose an accurate representation of the pdfs of the frequencies by mixtures of von Mises pdfs, which yields closed-form expectations. We define the algorithm VALSE in which the estimates of the pdfs and parameters are iteratively updated. VALSE is a gridless, convergent method, does not require parameter tuning, can easily include prior knowledge about the frequencies and provides approximate posterior pdfs based on which the uncertainty in line spectral estimation can be quantified. Simulation results show that accounting for the uncertainty of frequency estimates, rather than computing just point estimates, significantly improves the performance. The performance of VALSE is superior to that of state-of-the-art methods and closely approaches the Cram\'er-Rao bound computed for the true model order.Comment: 15 pages, 8 figures, accepted for publication in IEEE Transactions on Signal Processin
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