Results 1  10
of
155
Smooth minimization of nonsmooth functions
 Math. Programming
, 2005
"... In this paper we propose a new approach for constructing efficient schemes for nonsmooth convex optimization. It is based on a special smoothing technique, which can be applied to the functions with explicit maxstructure. Our approach can be considered as an alternative to blackbox minimization. F ..."
Abstract

Cited by 523 (1 self)
 Add to MetaCart
(Show Context)
In this paper we propose a new approach for constructing efficient schemes for nonsmooth convex optimization. It is based on a special smoothing technique, which can be applied to the functions with explicit maxstructure. Our approach can be considered as an alternative to blackbox minimization. From the viewpoint of efficiency estimates, we manage to improve the traditional bounds on the number of iterations of the gradient schemes from O unchanged. 1 ɛ 2 to O
A New Class of Upper Bounds on the Log Partition Function
 In Uncertainty in Artificial Intelligence
, 2002
"... Bounds on the log partition function are important in a variety of contexts, including approximate inference, model fitting, decision theory, and large deviations analysis [11, 5, 4]. We introduce a new class of upper bounds on the log partition function, based on convex combinations of distribution ..."
Abstract

Cited by 225 (32 self)
 Add to MetaCart
Bounds on the log partition function are important in a variety of contexts, including approximate inference, model fitting, decision theory, and large deviations analysis [11, 5, 4]. We introduce a new class of upper bounds on the log partition function, based on convex combinations of distributions in the exponential domain, that is applicable to an arbitrary undirected graphical model. In the special case of convex combinations of treestructured distributions, we obtain a family of variational problems, similar to the Bethe free energy, but distinguished by the following desirable properties: (i) they are convex, and have a unique global minimum; and (ii) the global minimum gives an upper bound on the log partition function. The global minimum is defined by stationary conditions very similar to those defining xed points of belief propagation (BP) or treebased reparameterization [see 13, 14]. As with BP fixed points, the elements of the minimizing argument can be used as approximations to the marginals of the original model. The analysis described here can be extended to structures of higher treewidth (e.g., hypertrees), thereby making connections with more advanced approximations (e.g., Kikuchi and variants [15, 10]).
DETERMINANT MAXIMIZATION WITH LINEAR MATRIX INEQUALITY CONSTRAINTS
"... The problem of maximizing the determinant of a matrix subject to linear matrix inequalities arises in many fields, including computational geometry, statistics, system identification, experiment design, and information and communication theory. It can also be considered as a generalization of the s ..."
Abstract

Cited by 223 (18 self)
 Add to MetaCart
The problem of maximizing the determinant of a matrix subject to linear matrix inequalities arises in many fields, including computational geometry, statistics, system identification, experiment design, and information and communication theory. It can also be considered as a generalization of the semidefinite programming problem. We give an overview of the applications of the determinant maximization problem, pointing out simple cases where specialized algorithms or analytical solutions are known. We then describe an interiorpoint method, with a simplified analysis of the worstcase complexity and numerical results that indicate that the method is very efficient, both in theory and in practice. Compared to existing specialized algorithms (where they are available), the interiorpoint method will generally be slower; the advantage is that it handles a much wider variety of problems.
MAP estimation via agreement on trees: Messagepassing and linear programming
, 2002
"... We develop and analyze methods for computing provably optimal maximum a posteriori (MAP) configurations for a subclass of Markov random fields defined on graphs with cycles. By decomposing the original distribution into a convex combination of treestructured distributions, we obtain an upper bound ..."
Abstract

Cited by 191 (9 self)
 Add to MetaCart
We develop and analyze methods for computing provably optimal maximum a posteriori (MAP) configurations for a subclass of Markov random fields defined on graphs with cycles. By decomposing the original distribution into a convex combination of treestructured distributions, we obtain an upper bound on the optimal value of the original problem (i.e., the log probability of the MAP assignment) in terms of the combined optimal values of the tree problems. We prove that this upper bound is tight if and only if all the tree distributions share an optimal configuration in common. An important implication is that any such shared configuration must also be a MAP configuration for the original distribution. Next we develop two approaches to attempting to obtain tight upper bounds: (a) a treerelaxed linear program (LP), which is derived from the Lagrangian dual of the upper bounds; and (b) a treereweighted maxproduct messagepassing algorithm that is related to but distinct from the maxproduct algorithm. In this way, we establish a connection between a certain LP relaxation of the modefinding problem, and a reweighted form of the maxproduct (minsum) messagepassing algorithm.
MAP estimation via agreement on (hyper)trees: Messagepassing and linear programming approaches
 IEEE Transactions on Information Theory
, 2002
"... We develop an approach for computing provably exact maximum a posteriori (MAP) configurations for a subclass of problems on graphs with cycles. By decomposing the original problem into a convex combination of treestructured problems, we obtain an upper bound on the optimal value of the original ..."
Abstract

Cited by 147 (10 self)
 Add to MetaCart
(Show Context)
We develop an approach for computing provably exact maximum a posteriori (MAP) configurations for a subclass of problems on graphs with cycles. By decomposing the original problem into a convex combination of treestructured problems, we obtain an upper bound on the optimal value of the original problem (i.e., the log probability of the MAP assignment) in terms of the combined optimal values of the tree problems. We prove that this upper bound is met with equality if and only if the tree problems share an optimal configuration in common. An important implication is that any such shared configuration must also be a MAP configuration for the original problem. Next we present and analyze two methods for attempting to obtain tight upper bounds: (a) a treereweighted messagepassing algorithm that is related to but distinct from the maxproduct (minsum) algorithm; and (b) a treerelaxed linear program (LP), which is derived from the Lagrangian dual of the upper bounds. Finally, we discuss the conditions that govern when the relaxation is tight, in which case the MAP configuration can be obtained. The analysis described here generalizes naturally to convex combinations of hypertreestructured distributions.
Templates for Convex Cone Problems with Applications to Sparse Signal Recovery
, 2010
"... This paper develops a general framework for solving a variety of convex cone problems that frequently arise in signal processing, machine learning, statistics, and other fields. The approach works as follows: first, determine a conic formulation of the problem; second, determine its dual; third, app ..."
Abstract

Cited by 122 (6 self)
 Add to MetaCart
This paper develops a general framework for solving a variety of convex cone problems that frequently arise in signal processing, machine learning, statistics, and other fields. The approach works as follows: first, determine a conic formulation of the problem; second, determine its dual; third, apply smoothing; and fourth, solve using an optimal firstorder method. A merit of this approach is its flexibility: for example, all compressed sensing problems can be solved via this approach. These include models with objective functionals such as the totalvariation norm, ‖W x‖1 where W is arbitrary, or a combination thereof. In addition, the paper also introduces a number of technical contributions such as a novel continuation scheme, a novel approach for controlling the step size, and some new results showing that the smooth and unsmoothed problems are sometimes formally equivalent. Combined with our framework, these lead to novel, stable and computationally efficient algorithms. For instance, our general implementation is competitive with stateoftheart methods for solving intensively studied problems such as the LASSO. Further, numerical experiments show that one can solve the Dantzig selector problem, for which no efficient largescale solvers exist, in a few hundred iterations. Finally, the paper is accompanied with a software release. This software is not a single, monolithic solver; rather, it is a suite of programs and routines designed to serve as building blocks for constructing complete algorithms. Keywords. Optimal firstorder methods, Nesterov’s accelerated descent algorithms, proximal algorithms, conic duality, smoothing by conjugation, the Dantzig selector, the LASSO, nuclearnorm minimization.
Estimating divergence functionals and the likelihood ratio by penalized convex risk minimization
 In Advances in Neural Information Processing Systems (NIPS
, 2007
"... by convex risk minimization ..."
(Show Context)
A frameletbased image inpainting algorithm
 Applied and Computational Harmonic Analysis
"... Abstract. Image inpainting is a fundamental problem in image processing and has many applications. Motivated by the recent tight frame based methods on image restoration in either the image or the transform domain, we propose an iterative tight frame algorithm for image inpainting. We consider the c ..."
Abstract

Cited by 87 (40 self)
 Add to MetaCart
(Show Context)
Abstract. Image inpainting is a fundamental problem in image processing and has many applications. Motivated by the recent tight frame based methods on image restoration in either the image or the transform domain, we propose an iterative tight frame algorithm for image inpainting. We consider the convergence of this frameletbased algorithm by interpreting it as an iteration for minimizing a special functional. The proof of the convergence is under the framework of convex analysis and optimization theory. We also discuss the relationship of our method with other waveletbased methods. Numerical experiments are given to illustrate the performance of the proposed algorithm. Key words. Tight frame, inpainting, convex analysis 1. Introduction. The problem of inpainting [2] occurs when part of the pixel data in a picture is missing or overwritten by other means. This arises for example in restoring ancient drawings, where a portion of the picture is missing or damaged due to aging or scratch; or when an image is transmitted through a noisy channel. The task of inpainting is to recover the missing region from the incomplete data observed. Ideally, the restored image should possess shapes and patterns consistent
High dimensional analysis of semidefinite relaxations for sparse principal component analysis
, 2008
"... Principal component analysis (PCA) is a classical method for dimensionality reduction based on extracting the dominant eigenvectors of the sample covariance matrix. However, PCA is well known to behave poorly in the “large p, small n ” setting, in which the problem dimension p is comparable to or la ..."
Abstract

Cited by 85 (5 self)
 Add to MetaCart
(Show Context)
Principal component analysis (PCA) is a classical method for dimensionality reduction based on extracting the dominant eigenvectors of the sample covariance matrix. However, PCA is well known to behave poorly in the “large p, small n ” setting, in which the problem dimension p is comparable to or larger than the sample size n. This paper studies PCA in this highdimensional regime, but under the additional assumption that the maximal eigenvector is sparse, say with at most k nonzero components. We analyze two computationally tractable methods for recovering the support of this maximal eigenvector: (a) a simple diagonal cutoff method, which transitions from success to failure as a function of the order parameter θdia(n, p, k) = n/[k 2 log(p − k)]; and (b) a more sophisticated semidefinite programming (SDP) relaxation, which succeeds once the order parameter θsdp(n, p, k) = n/[k log(p − k)] is larger than a critical threshold. Our results thus highlight an interesting tradeoff between computational and statistical efficiency in highdimensional inference.
Analysis of HalfQuadratic Minimization Methods for Signal and Image Recovery
, 2003
"... Abstract. We address the minimization of regularized convex cost functions which are customarily used for edgepreserving restoration and reconstruction of signals and images. In order to accelerate computation, the multiplicative and the additive halfquadratic reformulation of the original cost ..."
Abstract

Cited by 52 (8 self)
 Add to MetaCart
(Show Context)
Abstract. We address the minimization of regularized convex cost functions which are customarily used for edgepreserving restoration and reconstruction of signals and images. In order to accelerate computation, the multiplicative and the additive halfquadratic reformulation of the original costfunction have been pioneered in Geman and Reynolds [IEEE Trans. Pattern Anal. Machine Intelligence, 14 (1992), pp. 367–383] and Geman and Yang [IEEE Trans. Image Process., 4 (1995), pp. 932–946]. The alternate minimization of the resultant (augmented) costfunctions has a simple explicit form. The goal of this paper is to provide a systematic analysis of the convergence rate achieved by these methods. For the multiplicative and additive halfquadratic regularizations, we determine their upper bounds for their rootconvergence factors. The bound for the multiplicative form is seen to be always smaller than the bound for the additive form. Experiments show that the number of iterations required for convergence for the multiplicative form is always less than that for the additive form. However, the computational cost of each iteration is much higher for the multiplicative form than for the additive form. The global assessment is that minimization using the additive form of halfquadratic regularization is faster than using the multiplicative form. When the additive form is applicable, it is hence recommended. Extensive experiments demonstrate that in our MATLAB implementation, both methods are substantially faster (in terms of computational times) than the standard MATLAB Optimization Toolbox routines used in our comparison study.