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71
Learning the Kernel Matrix with SemiDefinite Programming
, 2002
"... Kernelbased learning algorithms work by embedding the data into a Euclidean space, and then searching for linear relations among the embedded data points. The embedding is performed implicitly, by specifying the inner products between each pair of points in the embedding space. This information ..."
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Cited by 583 (22 self)
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Kernelbased learning algorithms work by embedding the data into a Euclidean space, and then searching for linear relations among the embedded data points. The embedding is performed implicitly, by specifying the inner products between each pair of points in the embedding space. This information is contained in the socalled kernel matrix, a symmetric and positive definite matrix that encodes the relative positions of all points. Specifying this matrix amounts to specifying the geometry of the embedding space and inducing a notion of similarity in the input spaceclassical model selection problems in machine learning. In this paper we show how the kernel matrix can be learned from data via semidefinite programming (SDP) techniques. When applied
Multiple kernel learning, conic duality, and the SMO algorithm
 In Proceedings of the 21st International Conference on Machine Learning (ICML
, 2004
"... While classical kernelbased classifiers are based on a single kernel, in practice it is often desirable to base classifiers on combinations of multiple kernels. Lanckriet et al. (2004) considered conic combinations of kernel matrices for the support vector machine (SVM), and showed that the optimiz ..."
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Cited by 296 (29 self)
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While classical kernelbased classifiers are based on a single kernel, in practice it is often desirable to base classifiers on combinations of multiple kernels. Lanckriet et al. (2004) considered conic combinations of kernel matrices for the support vector machine (SVM), and showed that the optimization of the coefficients of such a combination reduces to a convex optimization problem known as a quadraticallyconstrained quadratic program (QCQP). Unfortunately, current convex optimization toolboxes can solve this problem only for a small number of kernels and a small number of data points; moreover, the sequential minimal optimization (SMO) techniques that are essential in largescale implementations of the SVM cannot be applied because the cost function is nondifferentiable. We propose a novel dual formulation of the QCQP as a secondorder cone programming problem, and show how to exploit the technique of MoreauYosida regularization to yield a formulation to which SMO techniques can be applied. We present experimental results that show that our SMObased algorithm is significantly more efficient than the generalpurpose interior point methods available in current optimization toolboxes. 1.
A robust minimax approach to classification
 JOURNAL OF MACHINE LEARNING RESEARCH
, 2002
"... When constructing a classifier, the probability of correct classification of future data points should be maximized. We consider a binary classification problem where the mean and covariance matrix of each class are assumed to be known. No further assumptions are made with respect to the classcondi ..."
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Cited by 72 (7 self)
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When constructing a classifier, the probability of correct classification of future data points should be maximized. We consider a binary classification problem where the mean and covariance matrix of each class are assumed to be known. No further assumptions are made with respect to the classconditional distributions. Misclassification probabilities are then controlled in a worstcase setting: that is, under all possible choices of classconditional densities with given mean and covariance matrix, we minimize the worstcase (maximum) probability of misclassification of future data points. For a linear decision boundary, this desideratum is translated in a very direct way into a (convex) second order cone optimization problem, with complexity similar to a support vector machine problem. The minimax problem can be interpreted geometrically as minimizing the maximum of the Mahalanobis distances to the two classes. We address the issue of robustness with respect to estimation errors (in the means and covariances of the
KNITRO: An integrated package for nonlinear optimization
 Large Scale Nonlinear Optimization, 35–59, 2006
, 2006
"... This paper describes Knitro 5.0, a Cpackage for nonlinear optimization that combines complementary approaches to nonlinear optimization to achieve robust performance over a wide range of application requirements. The package is designed for solving largescale, smooth nonlinear programming problems ..."
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Cited by 52 (3 self)
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This paper describes Knitro 5.0, a Cpackage for nonlinear optimization that combines complementary approaches to nonlinear optimization to achieve robust performance over a wide range of application requirements. The package is designed for solving largescale, smooth nonlinear programming problems, and it is also effective for the following special cases: unconstrained optimization, nonlinear systems of equations, least squares, and linear and quadratic programming. Various algorithmic options are available, including two interior methods and an activeset method. The package provides crossover techniques between algorithmic options as well as automatic selection of options and settings. 1
On implementing a primaldual interiorpoint method for conic quadratic optimization
 MATHEMATICAL PROGRAMMING SER. B
, 2000
"... Conic quadratic optimization is the problem of minimizing a linear function subject to the intersection of an affine set and the product of quadratic cones. The problem is a convex optimization problem and has numerous applications in engineering, economics, and other areas of science. Indeed, linea ..."
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Cited by 43 (5 self)
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Conic quadratic optimization is the problem of minimizing a linear function subject to the intersection of an affine set and the product of quadratic cones. The problem is a convex optimization problem and has numerous applications in engineering, economics, and other areas of science. Indeed, linear and convex quadratic optimization is a special case. Conic quadratic optimization problems can in theory be solved efficiently using interiorpoint methods. In particular it has been shown by Nesterov and Todd that primaldual interiorpoint methods developed for linear optimization can be generalized to the conic quadratic case while maintaining their efficiency. Therefore, based on the work of Nesterov and Todd, we discuss an implementation of a primaldual interiorpoint method for solution of largescale sparse conic quadratic optimization problems. The main features of the implementation are it is based on a homogeneous and selfdual model, handles the rotated quadratic cone directly, employs a Mehrotra type predictorcorrector
Extracting Shared Subspace for Multilabel Classification
"... Multilabel problems arise in various domains such as multitopic document categorization and protein function prediction. One natural way to deal with such problems is to construct a binary classifier for each label, resulting in a set of independent binary classification problems. Since the multipl ..."
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Cited by 31 (1 self)
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Multilabel problems arise in various domains such as multitopic document categorization and protein function prediction. One natural way to deal with such problems is to construct a binary classifier for each label, resulting in a set of independent binary classification problems. Since the multiple labels share the same input space, and the semantics conveyed by different labels are usually correlated, it is essential to exploit the correlation information contained in different labels. In this paper, we consider a general framework for extracting shared structures in multilabel classification. In this framework, a common subspace is assumed to be shared among multiple labels. We show that the optimal solution to the proposed formulation can be obtained by solving a generalized eigenvalue problem, though the problem is nonconvex. For highdimensional problems, direct computation of the solution is expensive, and we develop an efficient algorithm for this case. One appealing feature of the proposed framework is that it includes several wellknown algorithms as special cases, thus elucidating their intrinsic relationships. We have conducted extensive experiments on eleven multitopic web page categorization tasks, and results demonstrate the effectiveness of the proposed formulation in comparison with several representative algorithms.
Multiclass Discriminant Kernel Learning via Convex Programming
"... Regularized kernel discriminant analysis (RKDA) performs linear discriminant analysis in the feature space via the kernel trick. Its performance depends on the selection of kernels. In this paper, we consider the problem of multiple kernel learning (MKL) for RKDA, in which the optimal kernel matrix ..."
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Cited by 24 (0 self)
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Regularized kernel discriminant analysis (RKDA) performs linear discriminant analysis in the feature space via the kernel trick. Its performance depends on the selection of kernels. In this paper, we consider the problem of multiple kernel learning (MKL) for RKDA, in which the optimal kernel matrix is obtained as a linear combination of prespecified kernel matrices. We show that the kernel learning problem in RKDA can be formulated as convex programs. First, we show that this problem can be formulated as a semidefinite program (SDP). Based on the equivalence relationship between RKDA and least square problems in the binaryclass case, we propose a convex quadratically constrained quadratic programming (QCQP) formulation for kernel learning in RKDA. A semiinfinite linear programming (SILP) formulation is derived to further improve the efficiency. We extend these formulations to the multiclass case based on a key result established in this paper. That is, the multiclass RKDA kernel learning problem can be decomposed into a set of binaryclass kernel learning problems which are constrained to share a common kernel. Based on this decomposition property, SDP formulations are proposed for the multiclass case. Furthermore, it leads naturally to QCQP and SILP formulations. As the performance of RKDA depends on the regularization parameter, we show that this parameter can also be optimized in a joint framework with the kernel. Extensive experiments have been conducted and analyzed, and connections to other algorithms are discussed.
G.: Robust classification with interval data
, 2003
"... We consider a binary, linear classification problem in which the data points are assumed to be unknown, but bounded within given hyperrectangles, i.e., the covariates are bounded within intervals explicitly given for each data point separately. We address the problem of designing a robust classifie ..."
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Cited by 19 (0 self)
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We consider a binary, linear classification problem in which the data points are assumed to be unknown, but bounded within given hyperrectangles, i.e., the covariates are bounded within intervals explicitly given for each data point separately. We address the problem of designing a robust classifier in this setting by minimizing the worstcase value of a given loss function, over all possible choices of the data in these multidimensional intervals. We examine in detail the application of this methodology to three specific loss functions, arising in support vector machines, in logistic regression and in minimax probability machines. We show that in each case, the resulting problem is amenable to efficient interiorpoint algorithms for convex optimization. The methods tend to produce sparse classifiers, i.e., they induce many zero coefficients in the resulting weight vectors, and we provide some theoretical grounds for this property. After presenting possible extensions of this framework to handle label errors and other uncertainty models, we discuss in some detail our implementation, which exploits the potential sparsity or a more general property referred to as regularity, of the input matrices. 1
Bounded biharmonic weights for realtime deformation
 In SIGGRAPH’11
"... Object deformation with linear blending dominates practical use as the fastest approach for transforming raster images, vector graphics, geometric models and animated characters. Unfortunately, linear blending schemes for skeletons or cages are not always easy to use because they may require manual ..."
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Cited by 17 (2 self)
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Object deformation with linear blending dominates practical use as the fastest approach for transforming raster images, vector graphics, geometric models and animated characters. Unfortunately, linear blending schemes for skeletons or cages are not always easy to use because they may require manual weight painting or modeling closed polyhedral envelopes around objects. Our goal is to make the design and control of deformations simpler by allowing the user to work freely with the most convenient combination of handle types. We develop linear blending weights that produce smooth and intuitive deformations for points, bones and cages of arbitrary topology. Our weights, called bounded biharmonic weights, minimize the Laplacian energy subject to bound constraints. Doing so spreads the influences of the controls in a shapeaware and localized manner, even for objects with complex and concave boundaries. The variational weight optimization also makes it possible to customize the weights so that they preserve the shape of specified essential object features. We demonstrate successful use of our blending weights for realtime deformation of 2D and 3D shapes.