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22
Sparse Reconstruction by Separable Approximation
, 2008
"... Finding sparse approximate solutions to large underdetermined linear systems of equations is a common problem in signal/image processing and statistics. Basis pursuit, the least absolute shrinkage and selection operator (LASSO), waveletbased deconvolution and reconstruction, and compressed sensing ( ..."
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Cited by 185 (27 self)
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Finding sparse approximate solutions to large underdetermined linear systems of equations is a common problem in signal/image processing and statistics. Basis pursuit, the least absolute shrinkage and selection operator (LASSO), waveletbased deconvolution and reconstruction, and compressed sensing (CS) are a few wellknown areas in which problems of this type appear. One standard approach is to minimize an objective function that includes a quadratic (ℓ2) error term added to a sparsityinducing (usually ℓ1) regularization term. We present an algorithmic framework for the more general problem of minimizing the sum of a smooth convex function and a nonsmooth, possibly nonconvex regularizer. We propose iterative methods in which each step is obtained by solving an optimization subproblem involving a quadratic term with diagonal Hessian (which is therefore separable in the unknowns) plus the original sparsityinducing regularizer. Our approach is suitable for cases in which this subproblem can be solved much more rapidly than the original problem. In addition to solving the standard ℓ2 − ℓ1 case, our framework yields an efficient solution technique for other regularizers, such as an ℓ∞norm regularizer and groupseparable (GS) regularizers. It also generalizes immediately to the case in which the data is complex rather than real. Experiments with CS problems show that our approach is competitive with the fastest known methods for the standard ℓ2 − ℓ1 problem, as well as being efficient on problems with other separable regularization terms.
Joint Bayesian Model Selection and Estimation of Noisy Sinusoids via Reversible Jump MCMC
, 1999
"... In this paper, the problem of joint Bayesian model selection and parameter estimation for sinusoids in white Gaussian noise is addressed. An original Bayesian model is proposed that allows us to define a posterior distribution on the parameter space. All Bayesian inference is then based on this dist ..."
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Cited by 44 (3 self)
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In this paper, the problem of joint Bayesian model selection and parameter estimation for sinusoids in white Gaussian noise is addressed. An original Bayesian model is proposed that allows us to define a posterior distribution on the parameter space. All Bayesian inference is then based on this distribution. Unfortunately, a direct evaluation of this distribution and of its features, including posterior model probabilities, requires evaluation of some complicated highdimensional integrals. We develop an efficient stochastic algorithm based on reversible jump Markov chain Monte Carlo methods to perform the Bayesian computation. A convergence result for this algorithm is established. In simulation, it appears that the performance of detection based on posterior model probabilities outperforms conventional detection schemes.
Robust Full Bayesian Learning for Radial Basis Networks
, 2001
"... We propose a hierachical full Bayesian model for radial basis networks. This model treats the model dimension (number of neurons), model parameters,... ..."
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Cited by 25 (4 self)
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We propose a hierachical full Bayesian model for radial basis networks. This model treats the model dimension (number of neurons), model parameters,...
Activation detection in functional MRI using subspace modeling and maximum likelihood estimation
 IEEE Trans. Med. Imag
, 1999
"... Abstract — A statistical method for detecting activated pixels in functional MRI (fMRI) data is presented. In this method, the fMRI time series measured at each pixel is modeled as the sum of a response signal which arises due to the experimentally controlled activationbaseline pattern, a nuisance ..."
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Cited by 20 (3 self)
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Abstract — A statistical method for detecting activated pixels in functional MRI (fMRI) data is presented. In this method, the fMRI time series measured at each pixel is modeled as the sum of a response signal which arises due to the experimentally controlled activationbaseline pattern, a nuisance component representing effects of no interest, and Gaussian white noise. For periodic activationbaseline patterns, the response signal is modeled by a truncated Fourier series with a known fundamental frequency but unknown Fourier coefficients. The nuisance subspace is assumed to be unknown. A maximum likelihood estimate is derived for the component of the nuisance subspace which is orthogonal to the response signal subspace. An estimate for the order of the nuisance subspace is obtained from an information theoretic criterion. A statistical test is derived and shown to be the uniformly most powerful (UMP) test invariant to a group of transformations which are natural to the hypothesis testing problem. The maximal invariant statistic used in this test has an F distribution. The theoretical F distribution under the null hypothesis strongly concurred with the experimental frequency distribution obtained by performing null experiments in which the subjects did not perform any activation task. Application of the theory to motor activation and visual stimulation fMRI studies is presented. Index Terms — Brain, functional MRI, maximum likelihood estimation, statistical analysis.
Robust Full Bayesian Learning for Neural Networks
, 1999
"... In this paper, we propose a hierarchical full Bayesian model for neural networks. This model treats the model dimension (number of neurons), model parameters, regularisation parameters and noise parameters as random variables that need to be estimated. We develop a reversible jump Markov chain Monte ..."
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Cited by 12 (9 self)
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In this paper, we propose a hierarchical full Bayesian model for neural networks. This model treats the model dimension (number of neurons), model parameters, regularisation parameters and noise parameters as random variables that need to be estimated. We develop a reversible jump Markov chain Monte Carlo (MCMC) method to perform the necessary computations. We find that the results obtained using this method are not only better than the ones reported previously, but also appear to be robust with respect to the prior specification. In addition, we propose a novel and computationally efficient reversible jump MCMC simulated annealing algorithm to optimise neural networks. This algorithm enables us to maximise the joint posterior distribution of the network parameters and the number of basis function. It performs a global search in the joint space of the parameters and number of parameters, thereby surmounting the problem of local minima. We show that by calibrating the full hierarchical ...
Bayesian Methods for Neural Networks
, 1999
"... Summary The application of the Bayesian learning paradigm to neural networks results in a flexible and powerful nonlinear modelling framework that can be used for regression, density estimation, prediction and classification. Within this framework, all sources of uncertainty are expressed and meas ..."
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Cited by 10 (0 self)
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Summary The application of the Bayesian learning paradigm to neural networks results in a flexible and powerful nonlinear modelling framework that can be used for regression, density estimation, prediction and classification. Within this framework, all sources of uncertainty are expressed and measured by probabilities. This formulation allows for a probabilistic treatment of our a priori knowledge, domain specific knowledge, model selection schemes, parameter estimation methods and noise estimation techniques. Many researchers have contributed towards the development of the Bayesian learning approach for neural networks. This thesis advances this research by proposing several novel extensions in the areas of sequential learning, model selection, optimisation and convergence assessment. The first contribution is a regularisation strategy for sequential learning based on extended Kalman filtering and noise estimation via evidence maximisation. Using the expectation maximisation (EM) algorithm, a similar algorithm is derived for batch learning. Much of the thesis is, however, devoted to Monte Carlo simulation methods. A robust Bayesian method is proposed to estimate,
Gorokhov, “Positivedefinite Toeplitz completion in DOA estimation for nonuniform linear antenna arrays. II. Partially augmentable arrays
 IEEE Transactions on Signal Processing
, 1999
"... Abstract—This paper considers the problem of directionofarrival (DOA) estimation for multiple uncorrelated plane waves incident on socalled “fully augmentable ” sparse linear arrays. In situations where a decision is made on the number of existing signal sources (m) prior to the estimation stage, ..."
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Cited by 8 (2 self)
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Abstract—This paper considers the problem of directionofarrival (DOA) estimation for multiple uncorrelated plane waves incident on socalled “fully augmentable ” sparse linear arrays. In situations where a decision is made on the number of existing signal sources (m) prior to the estimation stage, we investigate the conditions under which DOA estimation accuracy is effective (in the maximumlikelihood sense). In the case wherem is less than the number of antenna sensors (M), a new approach called “MUSICmaximumentropy equalization ” is proposed to improve DOA estimation performance in the “preasymptotic region ” of finite sample size (N) and signaltonoise ratio. A fullsized positive definite (p.d.) Toeplitz matrix is constructed from theM M direct data covariance matrix, and then, alternating projections are applied to find a p.d. Toeplitz matrix with mvariate signal eigensubspace (“signal subspace truncation”). When mM, Cramér–Rao bound analysis suggests that the minimal useful sample size N is rather large, even for arbitrarily strong signals. It is demonstrated that the wellknown direct augmentation approach (DAA) cannot approach the accuracy of the corresponding Cramér–Rao bound, even asymptotically (as N!1) and, therefore, needs to be improved. We present a new estimation method whereby signal subspace truncation of the DAA augmented matrix is used for initialization and is followed by a local maximumlikelihood optimization routine. The accuracy of this method is demonstrated to be asymptotically optimal for the various superior scenarios (mM) presented. Index Terms — Direction of arrival estimation, nonuniformly spaced arrays, linear arrays, Toeplitz matrices.
Reversible Jump MCMC for Joint Detection and Estimation of Sources in Coloured Noise
 IEEE TRANSACTIONS ON SIGNAL PROCESSING
, 2000
"... This paper presents a novel Bayesian solution to the difficult problem of joint detection and estimation of sources impinging on a single array of sensors in spatially coloured noise with arbitrary covariance structure. Robustness to the noise covariance structure is achieved by integrating out the ..."
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Cited by 7 (3 self)
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This paper presents a novel Bayesian solution to the difficult problem of joint detection and estimation of sources impinging on a single array of sensors in spatially coloured noise with arbitrary covariance structure. Robustness to the noise covariance structure is achieved by integrating out the unknown covariance matrix in an appropriate posterior distribution. The proposed procedure uses the Reversible Jump Markov Chain Monte Carlo method to extract the desired model order and direction of arrival parameters. We show that the determination of model order is consistent provided a particular hyperparameter is within a specified range. Simulation results support the effectiveness of the method.
MAP model order selection rule for 2D sinusoids in white noise
 IEEE TRANS. SIGNAL PROCESS
, 2005
"... We consider the problem of jointly estimating the number as well as the parameters of twodimensional (2D) sinusoidal signals, observed in the presence of an additive white Gaussian noise field. Existing solutions to this problem are based on model order selection rules and are derived for the par ..."
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Cited by 5 (2 self)
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We consider the problem of jointly estimating the number as well as the parameters of twodimensional (2D) sinusoidal signals, observed in the presence of an additive white Gaussian noise field. Existing solutions to this problem are based on model order selection rules and are derived for the parallel onedimensional (1D) problem. These criteria are then adapted to the 2D problem using heuristic arguments. Employing asymptotic considerations, we derive a maximum a posteriori (MAP) model order selection criterion for jointly estimating the parameters of the 2D sinusoids and their number. The proposed model order selection rule is strongly consistent. As an example, the model order selection criterion is applied as a component in an algorithm for parametric estimation and synthesis of textured images.
A Shift InvarianceBased OrderSelection Technique for Exponential Data Modelling
"... Abstract—This paper presents a new subspacebased technique for automatic detection of the number of exponentially damped sinusoids. It consists in studying the shiftinvariance of the dominant subspace of the Hankel data matrix. No threshold setting and no penalization terms are necessary. This mod ..."
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Cited by 4 (0 self)
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Abstract—This paper presents a new subspacebased technique for automatic detection of the number of exponentially damped sinusoids. It consists in studying the shiftinvariance of the dominant subspace of the Hankel data matrix. No threshold setting and no penalization terms are necessary. This modelbased method, easy to implement, can be plugged into most subspacebased harmonic retrieval algorithms. Index Terms—Exponentially damped sinusoids, harmonic retrieval, order selection, singular value decomposition, subspace, total least squares. I.