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14
Efficient Variational Inference in LargeScale Bayesian Compressed Sensing
"... We study linear models under heavytailed priors from a probabilistic viewpoint. Instead of computing a single sparse most probable (MAP) solution as in standard deterministic approaches, the focus in the Bayesian compressed sensing framework shifts towards capturing the full posterior distribution ..."
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We study linear models under heavytailed priors from a probabilistic viewpoint. Instead of computing a single sparse most probable (MAP) solution as in standard deterministic approaches, the focus in the Bayesian compressed sensing framework shifts towards capturing the full posterior distribution on the latent variables, which allows quantifying the estimation uncertainty and learning model parameters using maximum likelihood. The exact posterior distribution under the sparse linear model is intractable and we concentrate on variational Bayesian techniques to approximate it. Repeatedly computing Gaussian variances turns out to be a key requisite and constitutes the main computational bottleneck in applying variational techniques in largescale problems. We leverage on the recently proposed PerturbandMAP algorithm for drawing exact samples from Gaussian Markov random fields (GMRF). The main technical contribution of our paper is to show that estimating Gaussian variances using a relatively small number of such efficiently drawn random samples is much more effective than alternative generalpurpose variance estimation techniques. By reducing the problem of variance estimation to standard optimization primitives, the resulting variational algorithms are fully scalable and parallelizable, allowing Bayesian computations in extremely largescale problems with the same memory and time complexity requirements as conventional point estimation techniques. We illustrate these ideas with experiments in image deblurring.
A Bayesian maxproduct EM algorithm for reconstructing structured sparse signals
 in Proc. Conf. Inform. Sci. Syst
, 2012
"... Part of the Signal Processing Commons The complete bibliographic information for this item can be found at ..."
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Part of the Signal Processing Commons The complete bibliographic information for this item can be found at
SparsityAware Learning and Compressed Sensing: An Overview
, 2014
"... The notion of regularization has been widely used as a tool to address a number of problems that are usually encountered in Machine Learning. Improving the performance of an estimator by shrinking the norm of the MVU estimator, guarding against overfitting, coping with illconditioning, provid ..."
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Cited by 1 (1 self)
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The notion of regularization has been widely used as a tool to address a number of problems that are usually encountered in Machine Learning. Improving the performance of an estimator by shrinking the norm of the MVU estimator, guarding against overfitting, coping with illconditioning, provid
Fast L0based Image Deconvolution with Variational Bayesian Inference and MajorizationMinimization
"... Abstract—In this paper, we propose a new waveletbased image deconvolution algorithm to restore blurred images based on a Gaussian scale mixture model within the variational Bayesian framework. Our sparsityregularized model approximates an l0 norm by reweighting an l2 norm iteratively. We derive a ..."
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Abstract—In this paper, we propose a new waveletbased image deconvolution algorithm to restore blurred images based on a Gaussian scale mixture model within the variational Bayesian framework. Our sparsityregularized model approximates an l0 norm by reweighting an l2 norm iteratively. We derive a hierarchial Bayesian estimation with the use of subband adaptive majorizationminimization which simplifies computation of the posterior distribution, and has been shown to find good solutions in the nonconvex search space. The proposed method is flexible enough to incorporate groupsparse optimization. I.
Compressive Sensing of Signals from a GMM with Sparse Precision Matrices
"... This paper is concerned with compressive sensing of signals drawn from a Gaussian mixture model (GMM) with sparse precision matrices. Previous work has shown: (i) a signal drawn from a given GMM can be perfectly reconstructed from r noisefree measurements if the (dominant) rank of each covariance m ..."
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This paper is concerned with compressive sensing of signals drawn from a Gaussian mixture model (GMM) with sparse precision matrices. Previous work has shown: (i) a signal drawn from a given GMM can be perfectly reconstructed from r noisefree measurements if the (dominant) rank of each covariance matrix is less than r; (ii) a sparse Gaussian graphical model can be efficiently estimated from fullyobserved training signals using graphical lasso. This paper addresses a problem more challenging than both (i) and (ii), by assuming that the GMM is unknown and each signal is only observed through incomplete linear measurements. Under these challenging assumptions, we develop a hierarchical Bayesian method to simultaneously estimate the GMM and recover the signals using solely the incomplete measurements and a Bayesian shrinkage prior that promotes sparsity of the Gaussian precision matrices. In addition, we provide theoretical performance bounds to relate the reconstruction error to the number of signals for which measurements are available, the sparsity level of precision matrices, and the “incompleteness” of measurements. The proposed method is demonstrated extensively on compressive sensing of imagery and video, and the results with simulated and hardwareacquired real measurements show significant performance improvement over stateoftheart methods.
Expectation maximization hard thresholding methods for sparse signal reconstruction
, 2011
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Sparse Recovery with Partial Support Knowledge
"... The goal of sparse recovery is to recover the (approximately) best ksparse approximation ˆx of an ndimensional vector x from linear measurements Ax of x. We consider a variant of the problem which takes into account partial knowledge about the signal. In particular, we focus on the scenario where ..."
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The goal of sparse recovery is to recover the (approximately) best ksparse approximation ˆx of an ndimensional vector x from linear measurements Ax of x. We consider a variant of the problem which takes into account partial knowledge about the signal. In particular, we focus on the scenario where, after the measurements are taken, we are given a set S of size s that is supposed to contain most of the “large ” coefficients of x. The goal is then to find ˆx such that ‖x − ˆx‖p ≤ C min ksparse x ′ supp(x ′)⊆S ‖x − x ′ ‖q. (1) We refer to this formulation as the sparse recovery with partial support knowledge problem (SRPSK). We show that SRPSK can be solved, up to an approximation factor of C = 1 + ɛ, using O((k/ɛ) log(s/k)) measurements, for p = q = 2. Moreover, this bound is tight as long as s = O(ɛn / log(n/ɛ)). This completely resolves the asymptotic measurement complexity of the problem except for a very small range of the parameter s. To the best of our knowledge, this is the first variant of (1+ɛ)approximate sparse recovery for which the asymptotic measurement complexity has been determined.
Research Article SelfAdaptive Image Reconstruction Inspired by Insect Compound Eye Mechanism
"... which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Inspired by the mechanism of imaging and adaptation to luminosity in insect compound eyes (ICE), we propose an ICEbased adaptive reconstruction method (ARMICE), which can adj ..."
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which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Inspired by the mechanism of imaging and adaptation to luminosity in insect compound eyes (ICE), we propose an ICEbased adaptive reconstruction method (ARMICE), which can adjust the sampling vision field of image according to the environment light intensity. The target scene can be compressive, sampled independently with multichannel through ARMICE. Meanwhile, ARMICE can regulate the visual field of sampling to control imaging according to the environment light intensity. Based on the compressed sensing joint sparse model (JSM1), we establish an information processing system of ARMICE. The simulation of a fourchannel ARMICE system shows that the new method improves the peak signaltonoise ratio (PSNR) and resolution of the reconstructed target scene under two different cases of light intensity. Furthermore, there is no distinct block effect in the result, and the edge of the reconstructed image is smoother than that obtained by the other two reconstruction methods in this work. 1.