Results 1  10
of
108
SDPT3  a MATLAB software package for semidefinite programming
 OPTIMIZATION METHODS AND SOFTWARE
, 1999
"... This software package is a Matlab implementation of infeasible pathfollowing algorithms for solving standard semidefinite programming (SDP) problems. Mehrotratype predictorcorrector variants are included. Analogous algorithms for the homogeneous formulation of the standard SDP problem are also imp ..."
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Cited by 214 (11 self)
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This software package is a Matlab implementation of infeasible pathfollowing algorithms for solving standard semidefinite programming (SDP) problems. Mehrotratype predictorcorrector variants are included. Analogous algorithms for the homogeneous formulation of the standard SDP problem are also implemented. Four types of search directions are available, namely, the AHO, HKM, NT, and GT directions. A few classes of SDP problems are included as well. Numerical results for these classes show that our algorithms are fairly efficient and robust on problems with dimensions of the order of a few hundreds.
Unsupervised Learning of Image Manifolds by Semidefinite Programming
, 2004
"... Can we detect low dimensional structure in high dimensional data sets of images and video? The problem of dimensionality reduction arises often in computer vision and pattern recognition. In this paper, we propose a new solution to this problem based on semidefinite programming. Our algorithm can be ..."
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Cited by 162 (9 self)
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Can we detect low dimensional structure in high dimensional data sets of images and video? The problem of dimensionality reduction arises often in computer vision and pattern recognition. In this paper, we propose a new solution to this problem based on semidefinite programming. Our algorithm can be used to analyze high dimensional data that lies on or near a low dimensional manifold. It overcomes certain limitations of previous work in manifold learning, such as Isomap and locally linear embedding. We illustrate the algorithm on easily visualized examples of curves and surfaces, as well as on actual images of faces, handwritten digits, and solid objects.
MaximumMargin Matrix Factorization
 Advances in Neural Information Processing Systems 17
, 2005
"... We present a novel approach to collaborative prediction, using lownorm instead of lowrank factorizations. The approach is inspired by, and has strong connections to, largemargin linear discrimination. We show how to learn lownorm factorizations by solving a semidefinite program, and discuss ..."
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Cited by 146 (16 self)
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We present a novel approach to collaborative prediction, using lownorm instead of lowrank factorizations. The approach is inspired by, and has strong connections to, largemargin linear discrimination. We show how to learn lownorm factorizations by solving a semidefinite program, and discuss generalization error bounds for them.
Solving semidefinitequadraticlinear programs using SDPT3
 MATHEMATICAL PROGRAMMING
, 2003
"... This paper discusses computational experiments with linear optimization problems involving semidefinite, quadratic, and linear cone constraints (SQLPs). Many test problems of this type are solved using a new release of SDPT3, a Matlab implementation of infeasible primaldual pathfollowing algorithm ..."
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Cited by 139 (18 self)
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This paper discusses computational experiments with linear optimization problems involving semidefinite, quadratic, and linear cone constraints (SQLPs). Many test problems of this type are solved using a new release of SDPT3, a Matlab implementation of infeasible primaldual pathfollowing algorithms. The software developed by the authors uses Mehrotratype predictorcorrector variants of interiorpoint methods and two types of search directions: the HKM and NT directions. A discussion of implementation details is provided and computational results on problems from the SDPLIB and DIMACS Challenge collections are reported.
Proving Program Invariance and Termination by Parametric Abstraction, Lagrangian Relaxation and Semidefinite Programming
 IN VMCAI’2005: VERIFICATION, MODEL CHECKING, AND ABSTRACT INTERPRETATION, VOLUME 3385 OF LNCS
, 2005
"... In order to verify semialgebraic programs, we automatize the Floyd/Naur/Hoare proof method. The main task is to automatically infer valid invariants and rank functions. First we express the program semantics in polynomial form. Then the unknown rank function and invariants are abstracted in parametr ..."
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Cited by 63 (1 self)
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In order to verify semialgebraic programs, we automatize the Floyd/Naur/Hoare proof method. The main task is to automatically infer valid invariants and rank functions. First we express the program semantics in polynomial form. Then the unknown rank function and invariants are abstracted in parametric form. The implication in the Floyd/Naur/Hoare verification conditions is handled by abstraction into numerical constraints by Lagrangian relaxation. The remaining universal quantification is handled by semidefinite programming relaxation. Finally the parameters are computed using semidefinite programming solvers. This new approach exploits the recent progress in the numerical resolution of linear or bilinear matrix inequalities by semidefinite programming using efficient polynomial primal/dual interior point methods generalizing those wellknown in linear programming to convex optimization. The framework is applied to invariance and termination proof of sequential, nondeterministic, concurrent, and fair parallel imperative polynomial programs and can easily be extended to other safety and liveness properties.
Nonlinear dimensionality reduction by semidefinite programming and kernel matrix factorization
 in Proceedings of the 10th International Workshop on Artificial Intelligence and Statistics
, 2005
"... We describe an algorithm for nonlinear dimensionality reduction based on semidefinite programming and kernel matrix factorization. The algorithm learns a kernel matrix for high dimensional data that lies on or near a low dimensional manifold. In earlier work, the kernel matrix was learned by maximiz ..."
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Cited by 48 (5 self)
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We describe an algorithm for nonlinear dimensionality reduction based on semidefinite programming and kernel matrix factorization. The algorithm learns a kernel matrix for high dimensional data that lies on or near a low dimensional manifold. In earlier work, the kernel matrix was learned by maximizing the variance in feature space while preserving the distances and angles between nearest neighbors. In this paper, adapting recent ideas from semisupervised learning on graphs, we show that the full kernel matrix can be very well approximated by a product of smaller matrices. Representing the kernel matrix in this way, we can reformulate the semidefinite program in terms of a much smaller submatrix of inner products between randomly chosen landmarks. The new framework leads to orderofmagnitude reductions in computation time and makes it possible to study much larger problems in manifold learning. 1
The fastest mixing Markov process on a graph and a connection to a maximum variance unfolding problem
 SIAM REVIEW
, 2006
"... We consider a Markov process on a connected graph, with edges labeled with transition rates between the adjacent vertices. The distribution of the Markov process converges to the uniform distribution at a rate determined by the second smallest eigenvalue λ2 of the Laplacian of the weighted graph. I ..."
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Cited by 42 (4 self)
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We consider a Markov process on a connected graph, with edges labeled with transition rates between the adjacent vertices. The distribution of the Markov process converges to the uniform distribution at a rate determined by the second smallest eigenvalue λ2 of the Laplacian of the weighted graph. In this paper we consider the problem of assigning transition rates to the edges so as to maximize λ2 subject to a linear constraint on the rates. This is the problem of finding the fastest mixing Markov process (FMMP) on the graph. We show that the FMMP problem is a convex optimization problem, which can in turn be expressed as a semidefinite program, and therefore effectively solved numerically. We formulate a dual of the FMMP problem and show that it has a natural geometric interpretation as a maximum variance unfolding (MVU) problem, i.e., the problem of choosing a set of points to be as far apart as possible, measured by their variance, while respecting local distance constraints. This MVU problem is closely related to a problem recently proposed by Weinberger and Saul as a method for “unfolding ” highdimensional data that lies on a lowdimensional manifold. The duality between the FMMP and MVU problems sheds light on both problems, and allows us to characterize and, in some cases, find optimal solutions.
Fast solvers and efficient implementations for distance metric learning
 In ICML
, 2008
"... In this paper we study how to improve nearest neighbor classification by learning a Mahalanobis distance metric. We build on a recently proposed framework for distance metric learning known as large margin nearest neighbor (LMNN) classification. Our paper makes three contributions. First, we describ ..."
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Cited by 41 (5 self)
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In this paper we study how to improve nearest neighbor classification by learning a Mahalanobis distance metric. We build on a recently proposed framework for distance metric learning known as large margin nearest neighbor (LMNN) classification. Our paper makes three contributions. First, we describe a highly efficient solver for the particular instance of semidefinite programming that arises in LMNN classification; our solver can handle problems with billions of large margin constraints in a few hours. Second, we show how to reduce both training and testing times using metric ball trees; the speedups from ball trees are further magnified by learning low dimensional representations of the input space. Third, we show how to learn different Mahalanobis distance metrics in different parts of the input space. For large data sets, the use of locally adaptive distance metrics leads to even lower error rates. 1.
Learning with Matrix Factorization
, 2004
"... Matrices that can be factored into a product of two simpler matrices can serve as a useful and often natural model in the analysis of tabulated or highdimensional data. Models based on matrix factorization (Factor Analysis, PCA) have been extensively used in statistical analysis and machine learning ..."
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Cited by 38 (4 self)
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Matrices that can be factored into a product of two simpler matrices can serve as a useful and often natural model in the analysis of tabulated or highdimensional data. Models based on matrix factorization (Factor Analysis, PCA) have been extensively used in statistical analysis and machine learning for over a century, with many new formulations and models suggested in recent
Action respecting embedding
 In Proceedings of the TwentySecond International Conference on Machine Learning
, 2005
"... ARE is a nonlinear dimensionality reduction technique for embedding observation trajectories, which captures state dynamics that traditional methods do not. The core of ARE is a semidefinite optimization with constraints requiring actions to be distancepreserving in the resulting embedding. Unfort ..."
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Cited by 33 (4 self)
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ARE is a nonlinear dimensionality reduction technique for embedding observation trajectories, which captures state dynamics that traditional methods do not. The core of ARE is a semidefinite optimization with constraints requiring actions to be distancepreserving in the resulting embedding. Unfortunately, these constraints are quadratic in number and nonlocal (making recent scaling tricks inapplicable). Consequently, the original formulation was limited to relatively small datasets. This paper describes two techniques to mitigate these issues. We first introduce an actionguided variant of Isomap. Although it alone does not produce actionrespecting manifolds, it can be used to seed conjugate gradient to implicitly solve the primal variable formulation of the ARE optimization. The optimization is not convex, but the ActionGuided Isomap provides an excellent seed often very close to the global minimum. The resulting Scalable ARE procedure gives similar results to original ARE, but can be applied to datasets an order of magnitude larger. 1