The Experts below are selected from a list of 16080 Experts worldwide ranked by ideXlab platform
Sylvia Serfaty - One of the best experts on this subject based on the ideXlab platform.
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from the ginzburg landau model to vortex lattice problems
Communications in Mathematical Physics, 2012Co-Authors: Etienne Sandier, Sylvia SerfatyAbstract:We introduce a “Coulombian renormalized energy” W which is a logarithmic type of interaction between points in the plane, computed by a “renormalization.” We prove various of its properties, such as the existence of Minimizers, and show in particular, using results from number theory, that among lattice configurations the triangular lattice is the unique Minimizer. Its minimization in general remains open.
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from the ginzburg landau model to vortex lattice problems
arXiv: Analysis of PDEs, 2010Co-Authors: Etienne Sandier, Sylvia SerfatyAbstract:We study Minimizers of the two-dimensional Ginzburg-Landau energy with applied magnetic field, between the first and second critical fields. In this regime, minimizing configurations exhibit densely packed hexagonal vortex lattices, called Abrikosov lattices. We derive, in some asymptotic regime, a limiting interaction energy between points in the plane, $W$, which we prove has to be minimized by limits of energy-minimizing configurations, once blown-up at a suitable scale. This is a next order effect compared to the mean-field type results we previously established. The limiting "Coulombian renormalized energy" $W$ is a logarithmic type of interaction, computed by a "renormalization," and we believe it should be rather ubiquitous. We study various of its properties, and show in particular, using results from number theory, that among lattice configurations the hexagonal lattice is the unique Minimizer, thus providing a first rigorous hint at the Abrikosov lattice. Its minimization in general remains open. The derivation of $W$ uses energy methods: the framework of $\Gamma$-convergence, and an abstract scheme for obtaining lower bounds for "2-scale energies" via the ergodic theorem.
Ramin Zabih - One of the best experts on this subject based on the ideXlab platform.
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What Energy Functions Can Be Minimized via Graph Cuts?
IEEE Transactions on Pattern Analysis and Machine Intelligence, 2004Co-Authors: Vladimir Kolmogorov, Ramin ZabihAbstract:In the last few years, several new algorithms based on graph cuts have been developed to solve energy minimization problems in computer vision. Each of these techniques constructs a graph such that the minimum cut on the graph also minimizes the energy. Yet, because these graph constructions are complex and highly specific to a particular energy function, graph cuts have seen limited application to date. In this paper, we give a characterization of the energy functions that can be minimized by graph cuts. Our results are restricted to functions of binary variables. However, our work generalizes many previous constructions and is easily applicable to vision problems that involve large numbers of labels, such as stereo, motion, image restoration, and scene reconstruction. We give a precise characterization of what energy functions can be minimized using graph cuts, among the energy functions that can be written as a sum of terms containing three or fewer binary variables. We also provide a general-purpose construction to minimize such an energy function. Finally, we give a necessary condition for any energy function of binary variables to be minimized by graph cuts. Researchers who are considering the use of graph cuts to optimize a particular energy function can use our results to determine if this is possible and then follow our construction to create the appropriate graph. A software implementation is freely available.
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what energy functions can be minimized via graph cuts
European Conference on Computer Vision, 2002Co-Authors: Vladimir Kolmogorov, Ramin ZabihAbstract:In the last few years, several new algorithms based on graph cuts have been developed to solve energy minimization problems in computer vision. Each of these techniques constructs a graph such that the minimum cut on the graph also minimizes the energy. Yet because these graph constructions are complex and highly specific to a particular energy function, graph cuts have seen limited application to date. In this paper we characterize the energy functions that can be minimized by graph cuts. Our results are restricted to energy functions with binary variables. However, our work generalizes many previous constructions, and is easily applicable to vision problems that involve large numbers of labels, such as stereo, motion, image restoration and scene reconstruction. We present three main results: a necessary condition for any energy function that can be minimized by graph cuts; a sufficient condition for energy functions that can be written as a sum of functions of up to three variables at a time; and a general-purpose construction to minimize such an energy function. Researchers who are considering the use of graph cuts to optimize a particular energy function can use our results to determine if this is possible, and then follow our construction to create the appropriate graph.
Etienne Sandier - One of the best experts on this subject based on the ideXlab platform.
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from the ginzburg landau model to vortex lattice problems
Communications in Mathematical Physics, 2012Co-Authors: Etienne Sandier, Sylvia SerfatyAbstract:We introduce a “Coulombian renormalized energy” W which is a logarithmic type of interaction between points in the plane, computed by a “renormalization.” We prove various of its properties, such as the existence of Minimizers, and show in particular, using results from number theory, that among lattice configurations the triangular lattice is the unique Minimizer. Its minimization in general remains open.
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from the ginzburg landau model to vortex lattice problems
arXiv: Analysis of PDEs, 2010Co-Authors: Etienne Sandier, Sylvia SerfatyAbstract:We study Minimizers of the two-dimensional Ginzburg-Landau energy with applied magnetic field, between the first and second critical fields. In this regime, minimizing configurations exhibit densely packed hexagonal vortex lattices, called Abrikosov lattices. We derive, in some asymptotic regime, a limiting interaction energy between points in the plane, $W$, which we prove has to be minimized by limits of energy-minimizing configurations, once blown-up at a suitable scale. This is a next order effect compared to the mean-field type results we previously established. The limiting "Coulombian renormalized energy" $W$ is a logarithmic type of interaction, computed by a "renormalization," and we believe it should be rather ubiquitous. We study various of its properties, and show in particular, using results from number theory, that among lattice configurations the hexagonal lattice is the unique Minimizer, thus providing a first rigorous hint at the Abrikosov lattice. Its minimization in general remains open. The derivation of $W$ uses energy methods: the framework of $\Gamma$-convergence, and an abstract scheme for obtaining lower bounds for "2-scale energies" via the ergodic theorem.
Vladimir Kolmogorov - One of the best experts on this subject based on the ideXlab platform.
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What Energy Functions Can Be Minimized via Graph Cuts?
IEEE Transactions on Pattern Analysis and Machine Intelligence, 2004Co-Authors: Vladimir Kolmogorov, Ramin ZabihAbstract:In the last few years, several new algorithms based on graph cuts have been developed to solve energy minimization problems in computer vision. Each of these techniques constructs a graph such that the minimum cut on the graph also minimizes the energy. Yet, because these graph constructions are complex and highly specific to a particular energy function, graph cuts have seen limited application to date. In this paper, we give a characterization of the energy functions that can be minimized by graph cuts. Our results are restricted to functions of binary variables. However, our work generalizes many previous constructions and is easily applicable to vision problems that involve large numbers of labels, such as stereo, motion, image restoration, and scene reconstruction. We give a precise characterization of what energy functions can be minimized using graph cuts, among the energy functions that can be written as a sum of terms containing three or fewer binary variables. We also provide a general-purpose construction to minimize such an energy function. Finally, we give a necessary condition for any energy function of binary variables to be minimized by graph cuts. Researchers who are considering the use of graph cuts to optimize a particular energy function can use our results to determine if this is possible and then follow our construction to create the appropriate graph. A software implementation is freely available.
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what energy functions can be minimized via graph cuts
European Conference on Computer Vision, 2002Co-Authors: Vladimir Kolmogorov, Ramin ZabihAbstract:In the last few years, several new algorithms based on graph cuts have been developed to solve energy minimization problems in computer vision. Each of these techniques constructs a graph such that the minimum cut on the graph also minimizes the energy. Yet because these graph constructions are complex and highly specific to a particular energy function, graph cuts have seen limited application to date. In this paper we characterize the energy functions that can be minimized by graph cuts. Our results are restricted to energy functions with binary variables. However, our work generalizes many previous constructions, and is easily applicable to vision problems that involve large numbers of labels, such as stereo, motion, image restoration and scene reconstruction. We present three main results: a necessary condition for any energy function that can be minimized by graph cuts; a sufficient condition for energy functions that can be written as a sum of functions of up to three variables at a time; and a general-purpose construction to minimize such an energy function. Researchers who are considering the use of graph cuts to optimize a particular energy function can use our results to determine if this is possible, and then follow our construction to create the appropriate graph.
Dacheng Tao - One of the best experts on this subject based on the ideXlab platform.
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On the Rates of Convergence from Surrogate Risk Minimizers to the Bayes Optimal Classifier.
arXiv: Machine Learning, 2018Co-Authors: Jingwei Zhang, Tongliang Liu, Dacheng TaoAbstract:We study the rates of convergence from empirical surrogate risk Minimizers to the Bayes optimal classifier. Specifically, we introduce the notion of \emph{consistency intensity} to characterize a surrogate loss function and exploit this notion to obtain the rate of convergence from an empirical surrogate risk Minimizer to the Bayes optimal classifier, enabling fair comparisons of the excess risks of different surrogate risk Minimizers. The main result of the paper has practical implications including (1) showing that hinge loss is superior to logistic and exponential loss in the sense that its empirical Minimizer converges faster to the Bayes optimal classifier and (2) guiding to modify surrogate loss functions to accelerate the convergence to the Bayes optimal classifier.
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on the rates of convergence from surrogate risk Minimizers to the bayes optimal classifier
IEEE Transactions on Neural Networks, 2018Co-Authors: Jingwei Zhang, Tongliang Liu, Dacheng TaoAbstract:In classification, the use of 0-1 loss is preferable since the Minimizer of 0-1 risk leads to the Bayes optimal classifier. However, due to the nonconvexity of 0-1 loss, this optimization problem is NP-hard. Therefore, many convex surrogate loss functions have been adopted. Previous works have shown that if a Bayes-risk consistent loss function is used as a surrogate, the Minimizer of the empirical surrogate risk can achieve the Bayes optimal classifier as the sample size tends to infinity. Nevertheless, the comparison of convergence rates of Minimizers of different empirical surrogate risks to the Bayes optimal classifier has rarely been studied. Which characterization of the surrogate loss determines its convergence rate to the Bayes optimal classifier? Can we modify the loss function to achieve a faster convergence rate? In this article, we study the convergence rates of empirical surrogate Minimizers to the Bayes optimal classifier. Specifically, we introduce the notions of consistency intensity and conductivity to characterize a surrogate loss function and exploit this notion to obtain the rate of convergence from an empirical surrogate risk Minimizer to the Bayes optimal classifier, enabling fair comparisons of the excess risks of different surrogate risk Minimizers. The main result of this article has practical implications including: 1) showing that hinge loss (SVM) is superior to logistic loss (Logistic regression) and exponential loss (Adaboost) in the sense that its empirical Minimizer converges faster to the Bayes optimal classifier and 2) guiding the design of new loss functions to speed up the convergence rate to the Bayes optimal classifier with a data-dependent loss correction method inspired by our theorems.
