Results 1 
7 of
7
Multivariate generalized gaussian distribution: Convexity and graphical models
 IEEE Transaction on Signal Processing
, 2013
"... Abstract—We consider covariance estimation in themultivariate generalized Gaussian distribution (MGGD) and elliptically symmetric (ES) distribution. The maximum likelihood optimization associated with this problem is nonconvex, yet it has been proved that its global solution can be often computed ..."
Abstract

Cited by 9 (3 self)
 Add to MetaCart
(Show Context)
Abstract—We consider covariance estimation in themultivariate generalized Gaussian distribution (MGGD) and elliptically symmetric (ES) distribution. The maximum likelihood optimization associated with this problem is nonconvex, yet it has been proved that its global solution can be often computed via simple fixed point iterations. Our first contribution is a new analysis of this likelihood based on geodesic convexity that requires weaker assumptions. Our second contribution is a generalized framework for structured covariance estimation under sparsity constraints. We show that the optimizations can be formulated as convex minimization as long the MGGD shape parameter is larger than half and the sparsity pattern is chordal. These include, for example, maximum likelihood estimation of banded inverse covariances in multivariate Laplace distributions, which are associated with time varying autoregressive processes. Index Terms—Cholesky decomposition, geodesic convexity, graphical models, multivariate generalized Gaussian distribution. I.
Learning the Structure for Structured Sparsity
"... Abstract—Structured sparsity has recently emerged in statistics, machine learning and signal processing as a promising paradigm for learning in highdimensional settings. All existing methods for learning under the assumption of structured sparsity rely on prior knowledge on how to weight (or how to ..."
Abstract
 Add to MetaCart
(Show Context)
Abstract—Structured sparsity has recently emerged in statistics, machine learning and signal processing as a promising paradigm for learning in highdimensional settings. All existing methods for learning under the assumption of structured sparsity rely on prior knowledge on how to weight (or how to penalize) individual subsets of variables during the subset selection process, which is not available in general. Inferring group weights from data is a key open research problem in structured sparsity. In this paper, we propose a Bayesian approach to the problem of group weight learning. We model the group weights as hyperparameters of heavytailed priors on groups of variables and derive an approximate inference scheme to infer these hyperparameters. We empirically show that we are able to recover the model hyperparameters when the data are generated from the model, and we demonstrate the utility of learning weights in synthetic and real denoising problems. Index Terms — Structured sparsity, probabilistic modeling, Bayesian statistics, superGaussian prior, Gaussian scale mixture,
A MULTIVARIATE STATISTICAL MODEL FOR MULTIPLE IMAGES ACQUIRED BY HOMOGENEOUS OR HETEROGENEOUS SENSORS
"... This paper introduces a new statistical model for homogeneous images acquired by the same kind of sensor (e.g., two optical images) and heterogeneous images acquired by different sensors (e.g., optical and synthetic aperture radar (SAR) images). The proposed model assumes that each image pixel is d ..."
Abstract
 Add to MetaCart
(Show Context)
This paper introduces a new statistical model for homogeneous images acquired by the same kind of sensor (e.g., two optical images) and heterogeneous images acquired by different sensors (e.g., optical and synthetic aperture radar (SAR) images). The proposed model assumes that each image pixel is distributed according to a mixture of multidimensional distributions depending on the noise properties and on the transformation between the actual scene and the image intensities. The parameters of this new model can be estimated by the classical expectationmaximization algorithm. The estimated parameters are finally used to learn the relationships between the different images. This information can be used in many image processing applications, particularly those requiring a similarity measure (e.g., change detection or registration). Simulation results on synthetic and real images show the potential of the proposed model. A brief application to change detection between optical and SAR images is finally investigated. Index Terms — Image analysis, change detection, remote sensing, multitemporal images, mixture models, optical images, synthetic aperture radar.
Learning to Learn for Structured Sparsity
, 2014
"... Structured sparsity has recently emerged in statistics, machine learning and signal processing as a promising paradigm for learning in highdimensional settings. A number of methods have been proposed for learning under the assumption of structured sparsity, including group LASSO and graph LASSO. A ..."
Abstract
 Add to MetaCart
(Show Context)
Structured sparsity has recently emerged in statistics, machine learning and signal processing as a promising paradigm for learning in highdimensional settings. A number of methods have been proposed for learning under the assumption of structured sparsity, including group LASSO and graph LASSO. All of these methods rely on prior knowledge on how to weight (equivalently, how to penalize) individual subsets of variables during the subset selection process. However, these weights on groups of variables are in general unknown. Inferring group weights from data is a key open problem in research on structured sparsity. In this paper, we propose a Bayesian approach to the problem of group weight learning. We model the group weights as hyperparameters of heavytailed priors on groups of variables and derive an approximate inference scheme to infer these hyperparameters. We empirically show that we are able to recover the model hyperparameters when the data are generated from the model, and moreover, we demonstrate the utility of learning group weights in synthetic and real denoising problems. 1
SUBMITTED TO IEEE TRANS. ON SIGNAL PROCESSING 1 Generalized
"... robust shrinkage estimator and its application to STAP detection problem ..."
1Multivariate Generalized Gaussian Distribution: Convexity and Graphical Models
"... Abstract—We consider covariance estimation in the multivariate generalized Gaussian distribution (MGGD) and elliptically symmetric (ES) distribution. The maximum likelihood optimization associated with this problem is nonconvex, yet it has been proved that its global solution can be often compute ..."
Abstract
 Add to MetaCart
Abstract—We consider covariance estimation in the multivariate generalized Gaussian distribution (MGGD) and elliptically symmetric (ES) distribution. The maximum likelihood optimization associated with this problem is nonconvex, yet it has been proved that its global solution can be often computed via simple fixed point iterations. Our first contribution is a new analysis of this likelihood based on geodesic convexity that requires weaker assumptions. Our second contribution is a generalized framework for structured covariance estimation under sparsity constraints. We show that the optimizations can be formulated as convex minimization as long the MGGD shape parameter is larger than half and the sparsity pattern is chordal. These include, for example, maximum likelihood estimation of banded inverse covariances in multivariate Laplace distributions, which are associated with time varying autoregressive processes. Index Terms—Multivariate generalized Gaussian distribution, geodesic convexity, graphical models, Cholesky decomposition.
Applied Statistical Modeling of ThreeDimensional Natural Scene Data
, 2014
"... Copyright by ..."
(Show Context)