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40
Bayesian Compressive Sensing
, 2007
"... The data of interest are assumed to be represented as Ndimensional real vectors, and these vectors are compressible in some linear basis B, implying that the signal can be reconstructed accurately using only a small number M ≪ N of basisfunction coefficients associated with B. Compressive sensing ..."
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Cited by 141 (15 self)
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The data of interest are assumed to be represented as Ndimensional real vectors, and these vectors are compressible in some linear basis B, implying that the signal can be reconstructed accurately using only a small number M ≪ N of basisfunction coefficients associated with B. Compressive sensing is a framework whereby one does not measure one of the aforementioned Ndimensional signals directly, but rather a set of related measurements, with the new measurements a linear combination of the original underlying Ndimensional signal. The number of required compressivesensing measurements is typically much smaller than N, offering the potential to simplify the sensing system. Let f denote the unknown underlying Ndimensional signal, and g a vector of compressivesensing measurements, then one may approximate f accurately by utilizing knowledge of the (underdetermined) linear relationship between f and g, in addition to knowledge of the fact that f is compressible in B. In this paper we employ a Bayesian formalism for estimating the underlying signal f based on compressivesensing measurements g. The proposed framework has the following properties: (i) in addition to estimating the underlying signal f, “error bars ” are also estimated, these giving a measure of confidence in the inverted signal; (ii) using knowledge of the error bars, a principled means is provided for determining when a sufficient
Discriminative fields for modeling spatial dependencies in natural images
 In NIPS
, 2003
"... In this paper we present Discriminative Random Fields (DRF), a discriminative framework for the classification of natural image regions by incorporating neighborhood spatial dependencies in the labels as well as the observed data. The proposed model exploits local discriminative models and allows to ..."
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Cited by 108 (3 self)
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In this paper we present Discriminative Random Fields (DRF), a discriminative framework for the classification of natural image regions by incorporating neighborhood spatial dependencies in the labels as well as the observed data. The proposed model exploits local discriminative models and allows to relax the assumption of conditional independence of the observed data given the labels, commonly used in the Markov Random Field (MRF) framework. The parameters of the DRF model are learned using penalized maximum pseudolikelihood method. Furthermore, the form of the DRF model allows the MAP inference for binary classification problems using the graph mincut algorithms. The performance of the model was verified on the synthetic as well as the realworld images. The DRF model outperforms the MRF model in the experiments. 1
Adaptive Sparseness for Supervised Learning
 IEEE Transactions on Pattern Analysis and Machine Intelligence
, 2003
"... The goal of supervised learning is to infer a functional mapping based on a set of training examples. To achieve good generalization, it is necessary to control the "complexity" of the learned function. In Bayesian approaches, this is done by adopting a prior for the parameters of the fu ..."
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Cited by 83 (4 self)
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The goal of supervised learning is to infer a functional mapping based on a set of training examples. To achieve good generalization, it is necessary to control the "complexity" of the learned function. In Bayesian approaches, this is done by adopting a prior for the parameters of the function being learned. We propose a Bayesian approach to supervised learning, which leads to sparse solutions; that is, in which irrelevant parameters are automatically set exactly to zero. Other ways to obtain sparse classifiers (such as Laplacian priors, support vector machines) involve (hyper)parameters which control the degree of sparseness of the resulting classifiers; these parameters have to be somehow adjusted/estimated from the training data. In contrast, our approach does not involve any (hyper)parameters to be adjusted or estimated. This is achieved by a hierarchicalBayes interpretation of the Laplacian prior, which is then modified by the adoption of a Jeffreys' noninformative hyperprior. Implementation is carried out by an expectationmaximization (EM) algorithm. Experiments with several benchmark data sets show that the proposed approach yields stateoftheart performance. In particular, our method outperforms SVMs and performs competitively with the best alternative techniques, although it involves no tuning or adjustment of sparsenesscontrolling hyperparameters.
Sparse Representation For Computer Vision and Pattern Recognition
, 2009
"... Techniques from sparse signal representation are beginning to see significant impact in computer vision, often on nontraditional applications where the goal is not just to obtain a compact highfidelity representation of the observed signal, but also to extract semantic information. The choice of ..."
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Cited by 48 (1 self)
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Techniques from sparse signal representation are beginning to see significant impact in computer vision, often on nontraditional applications where the goal is not just to obtain a compact highfidelity representation of the observed signal, but also to extract semantic information. The choice of dictionary plays a key role in bridging this gap: unconventional dictionaries consisting of, or learned from, the training samples themselves provide the key to obtaining stateoftheart results and to attaching semantic meaning to sparse signal representations. Understanding the good performance of such unconventional dictionaries in turn demands new algorithmic and analytical techniques. This review paper highlights a few representative examples of how the interaction between sparse signal representation and computer vision can enrich both fields, and raises a number of open questions for further study.
A new view of automatic relevance determination
 In NIPS 20
, 2008
"... Automatic relevance determination (ARD) and the closelyrelated sparse Bayesian learning (SBL) framework are effective tools for pruning large numbers of irrelevant features leading to a sparse explanatory subset. However, popular update rules used for ARD are either difficult to extend to more gene ..."
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Cited by 41 (8 self)
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Automatic relevance determination (ARD) and the closelyrelated sparse Bayesian learning (SBL) framework are effective tools for pruning large numbers of irrelevant features leading to a sparse explanatory subset. However, popular update rules used for ARD are either difficult to extend to more general problems of interest or are characterized by nonideal convergence properties. Moreover, it remains unclear exactly how ARD relates to more traditional MAP estimationbased methods for learning sparse representations (e.g., the Lasso). This paper furnishes an alternative means of expressing the ARD cost function using auxiliary functions that naturally addresses both of these issues. First, the proposed reformulation of ARD can naturally be optimized by solving a series of reweighted ℓ1 problems. The result is an efficient, extensible algorithm that can be implemented using standard convex programming toolboxes and is guaranteed to converge to a local minimum (or saddle point). Secondly, the analysis reveals that ARD is exactly equivalent to performing standard MAP estimation in weight space using a particular feature and noisedependent, nonfactorial weight prior. We then demonstrate that this implicit prior maintains several desirable advantages over conventional priors with respect to feature selection. Overall these results suggest alternative cost functions and update procedures for selecting features and promoting sparse solutions in a variety of general situations. In particular, the methodology readily extends to handle problems such as nonnegative sparse coding and covariance component estimation. 1
An empirical bayesian strategy for solving the simultaneous sparse approximation problem
 IEEE Trans. Sig. Proc
, 2007
"... Abstract—Given a large overcomplete dictionary of basis vectors, the goal is to simultaneously represent 1 signal vectors using coefficient expansions marked by a common sparsity profile. This generalizes the standard sparse representation problem to the case where multiple responses exist that were ..."
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Cited by 41 (8 self)
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Abstract—Given a large overcomplete dictionary of basis vectors, the goal is to simultaneously represent 1 signal vectors using coefficient expansions marked by a common sparsity profile. This generalizes the standard sparse representation problem to the case where multiple responses exist that were putatively generated by the same small subset of features. Ideally, the associated sparse generating weights should be recovered, which can have physical significance in many applications (e.g., source localization). The generic solution to this problem is intractable and, therefore, approximate procedures are sought. Based on the concept of automatic relevance determination, this paper uses an empirical Bayesian prior to estimate a convenient posterior distribution over candidate basis vectors. This particular approximation enforces a common sparsity profile and consistently places its prominent posterior mass on the appropriate region of weightspace necessary for simultaneous sparse recovery. The resultant algorithm is then compared with multiple response extensions of matching pursuit, basis pursuit, FOCUSS, and Jeffreys priorbased Bayesian methods, finding that it often outperforms the others. Additional motivation for this particular choice of cost function is also provided, including the analysis of global and local minima and a variational derivation that highlights the similarities and differences between the proposed algorithm and previous approaches. Index Terms—Automatic relevance determination, empirical Bayes, multiple response models, simultaneous sparse approximation, sparse Bayesian learning, variable selection. I.
On semisupervised classification
 In
, 2005
"... A graphbased prior is proposed for parametric semisupervised classification. The prior utilizes both labelled and unlabelled data; it also integrates features from multiple views of a given sample (e.g., multiple sensors), thus implementing a Bayesian form of cotraining. An EM algorithm for train ..."
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Cited by 39 (8 self)
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A graphbased prior is proposed for parametric semisupervised classification. The prior utilizes both labelled and unlabelled data; it also integrates features from multiple views of a given sample (e.g., multiple sensors), thus implementing a Bayesian form of cotraining. An EM algorithm for training the classifier automatically adjusts the tradeoff between the contributions of: (a) the labelled data; (b) the unlabelled data; and (c) the cotraining information. Active label query selection is performed using a mutual information based criterion that explicitly uses the unlabelled data and the cotraining information. Encouraging results are presented on public benchmarks and on measured data from single and multiple sensors. 1
Variational EM algorithms for nonGaussian latent variable models
 In Advances in Neural Information Processing Systems 18
, 2006
"... We consider criteria for variational representations of nonGaussian latent variables, and derive variational EM algorithms in general form. We establish a general equivalence among convex bounding methods, evidence based methods, and ensemble learning/Variational Bayes methods, which has previous ..."
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Cited by 38 (13 self)
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We consider criteria for variational representations of nonGaussian latent variables, and derive variational EM algorithms in general form. We establish a general equivalence among convex bounding methods, evidence based methods, and ensemble learning/Variational Bayes methods, which has previously been demonstrated only for particular cases. 1
A unified Bayesian framework for MEG/EEG source imaging
 Neuroimage
, 2009
"... The illposed nature of the MEG (or related EEG) source localization problem requires the incorporation of prior assumptions when choosing an appropriate solution out of an infinite set of candidates. Bayesian approaches are useful in this capacity because they allow these assumptions to be explicit ..."
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Cited by 26 (2 self)
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The illposed nature of the MEG (or related EEG) source localization problem requires the incorporation of prior assumptions when choosing an appropriate solution out of an infinite set of candidates. Bayesian approaches are useful in this capacity because they allow these assumptions to be explicitly quantified using postulated prior distributions. However, the means by which these priors are chosen, as well as the estimation and inference procedures that are subsequently adopted to affect localization, have led to a daunting array of algorithms with seemingly very different properties and assumptions. From the vantage point of a simple Gaussian scale mixture model with flexible covariance components, this paper analyzes and extends several broad categories of Bayesian inference directly applicable to source localization including empirical Bayesian approaches, standard MAP estimation, and multiple variational Bayesian (VB) approximations. Theoretical properties related to convergence, global and local minima, and localization bias are analyzed and fast algorithms are derived that improve upon existing methods. This perspective leads to explicit connections between many established algorithms and suggests natural extensions for handling unknown dipole orientations, extended source configurations, correlated sources, temporal smoothness, and computational expediency. Specific imaging methods elucidated under this paradigm include weighted minimum ℓ2norm, FOCUSS, MCE, VESTAL, sLORETA, ReML and covariance component estimation, beamforming, variational Bayes, the Laplace approximation, and automatic relevance determination (ARD). Perhaps surprisingly, all of these methods can be formulated as particular cases of covariance component estimation using different concave regularization terms and optimization rules, making general theoretical analyses and algorithmic extensions/improvements particularly relevant. I.
Bayesian Hyperspectral Image Segmentation with Discriminative Class Learning
"... Abstract. This paper presents a new Bayesian approach to hyperspectral image segmentation that boosts the performance of the discriminative classifiers. This is achieved by combining class densities based on discriminative classifiers with a MultiLevel Logistic MarkovGibs prior. This density favor ..."
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Cited by 17 (9 self)
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Abstract. This paper presents a new Bayesian approach to hyperspectral image segmentation that boosts the performance of the discriminative classifiers. This is achieved by combining class densities based on discriminative classifiers with a MultiLevel Logistic MarkovGibs prior. This density favors neighbouring labels of the same class. The adopted discriminative classifier is the Fast Sparse Multinomial Regression. The discrete optimization problem one is led to is solved efficiently via graph cut tools. The effectiveness of the proposed method is evaluated, with simulated and real AVIRIS images, in two directions: 1) to improve the classification performance and 2) to decrease the size of the training sets. 1