The Experts below are selected from a list of 13191 Experts worldwide ranked by ideXlab platform
Luca Baracco - One of the best experts on this subject based on the ideXlab platform.
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The range of the tangential Cauchy–Riemann system to a CR Embedded Manifold
Inventiones mathematicae, 2012Co-Authors: Luca BaraccoAbstract:We prove that every compact, pseudoconvex, orientable, CR Manifold of ${\Bbb{C}}^{n}$ , bounds a complex Manifold in the C ^∞ sense. In particular, $\bar{\partial}_{b}$ has closed range.
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Erratum to: The range of the tangential Cauchy–Riemann system to a CR Embedded Manifold
Inventiones mathematicae, 2012Co-Authors: Luca BaraccoAbstract:We prove that every compact, pseudoconvex, orientable, CR Manifold of $\C^n$, bounds a complex Manifold in the $C^\infty$ sense. In particular, the tangential Cauchy-Riemann system has closed range.
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the range of the tangential cauchy riemann system to a cr Embedded Manifold
Inventiones Mathematicae, 2012Co-Authors: Luca BaraccoAbstract:We prove that every compact, pseudoconvex, orientable, CR Manifold of \({\Bbb{C}}^{n}\), bounds a complex Manifold in the C∞ sense. In particular, \(\bar{\partial}_{b}\) has closed range.
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the range of the tangential cauchy riemann system on a cr Embedded Manifold
arXiv: Complex Variables, 2011Co-Authors: Luca BaraccoAbstract:We prove that every compact, pseudoconvex, orientable, CR Manifold of $\C^n$, bounds a complex Manifold in the $C^\infty$ sense. In particular, the tangential Cauchy-Riemann system has closed range.
Gal Chechik - One of the best experts on this subject based on the ideXlab platform.
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Online learning in the Embedded Manifold of low-rank matrices
Journal of Machine Learning Research, 2012Co-Authors: Uri Shalit, Daphna Weinshall, Gal ChechikAbstract:When learning models that are represented in matrix forms, enforcing a low-rank constraint can dramatically improve the memory and run time complexity, while providing a natural regularization of the model. However, naive approaches to minimizing functions over the set of low-rank matrices are either prohibitively time consuming (repeated singular value decomposition of the matrix) or numerically unstable (optimizing a factored representation of the low-rank matrix). We build on recent advances in optimization over Manifolds, and describe an iterative online learning procedure, consisting of a gradient step, followed by a second-order retraction back to the Manifold. While the ideal retraction is costly to compute, and so is the projection operator that approximates it, we describe another retraction that can be computed efficiently. It has run time and memory complexity of O((n+m)k) for a rank-k matrix of dimension m×n, when using an online procedure with rank-one gradients. We use this algorithm, LORETA, to learn a matrix-form similarity measure over pairs of documents represented as high dimensional vectors. LORETA improves the mean average precision over a passive-aggressive approach in a factorized model, and also improves over a full model trained on pre-selected features using the same memory requirements. We further adapt LORETA to learn positive semi-definite low-rank matrices, providing an online algorithm for low-rank metric learning. LORETA also shows consistent improvement over standard weakly supervised methods in a large (1600 classes and 1 million images, using ImageNet) multilabel image classification task.
Chechikgal - One of the best experts on this subject based on the ideXlab platform.
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Online learning in the Embedded Manifold of low-rank matrices
Journal of Machine Learning Research, 2012Co-Authors: Shalituri, Weinshalldaphna, ChechikgalAbstract:When learning models that are represented in matrix forms, enforcing a low-rank constraint can dramatically improve the memory and run time complexity, while providing a natural regularization of t...
Uri Shalit - One of the best experts on this subject based on the ideXlab platform.
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Online learning in the Embedded Manifold of low-rank matrices
Journal of Machine Learning Research, 2012Co-Authors: Uri Shalit, Daphna Weinshall, Gal ChechikAbstract:When learning models that are represented in matrix forms, enforcing a low-rank constraint can dramatically improve the memory and run time complexity, while providing a natural regularization of the model. However, naive approaches to minimizing functions over the set of low-rank matrices are either prohibitively time consuming (repeated singular value decomposition of the matrix) or numerically unstable (optimizing a factored representation of the low-rank matrix). We build on recent advances in optimization over Manifolds, and describe an iterative online learning procedure, consisting of a gradient step, followed by a second-order retraction back to the Manifold. While the ideal retraction is costly to compute, and so is the projection operator that approximates it, we describe another retraction that can be computed efficiently. It has run time and memory complexity of O((n+m)k) for a rank-k matrix of dimension m×n, when using an online procedure with rank-one gradients. We use this algorithm, LORETA, to learn a matrix-form similarity measure over pairs of documents represented as high dimensional vectors. LORETA improves the mean average precision over a passive-aggressive approach in a factorized model, and also improves over a full model trained on pre-selected features using the same memory requirements. We further adapt LORETA to learn positive semi-definite low-rank matrices, providing an online algorithm for low-rank metric learning. LORETA also shows consistent improvement over standard weakly supervised methods in a large (1600 classes and 1 million images, using ImageNet) multilabel image classification task.
Qiansheng Cheng - One of the best experts on this subject based on the ideXlab platform.
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Nonlinear Discriminant Analysis on Embedded Manifold
IEEE Transactions on Circuits and Systems for Video Technology, 2007Co-Authors: Shuicheng Yan, Hongjiang Zhang, Benyu Zhang, Qiansheng ChengAbstract:Traditional Manifold learning algorithms, such as ISOMAP, LLE, and Laplacian Eigenmap, mainly focus on uncovering the latent low-dimensional geometry structure of the training samples in an unsupervised manner where useful class information is ignored. Therefore, the derived low-dimensional representations are not necessarily optimal in discriminative capability. In this paper, we study the discriminant analysis problem by considering the nonlinear Manifold structure of data space. To this end, firstly, a new clustering algorithm, called Intra-Cluster Balanced K-Means (ICBKM), is proposed to partition the samples into multiple clusters while ensure that there are balanced samples for the classes within each cluster; approximately, each cluster can be considered as a local patch on the Embedded Manifold. Then, the local discriminative projections for different clusters are simultaneously calculated by optimizing the global Fisher Criterion based on the cluster weighted data representation. Compared with traditional linear/kernel discriminant analysis (KDA) algorithms, our proposed algorithm has the following characteristics: 1) it essentially is a KDA algorithm with specific geometry-adaptive-kernel tailored to the specific data structure, in contrast to traditional KDA in which the kernel is fixed and independent to the data set; 2) it is approximately a locally linear while globally nonlinear discriminant analyzer; 3) it does not need to store the original samples for computing the low-dimensional representation of a new data; and 4) it is computationally efficient compared with traditional KDA when the sample number is large. The toy problem on artificial data demonstrates the effectiveness of our proposed algorithm in deriving discriminative representations for problems with nonlinear classification hyperplane. The face recognition experiments on YALE and CMU PIE databases show that our proposed algorithm significantly outperforms linear discriminant analysis (LDA) as well as Mixture LDA, and has higher accuracy than KDA with traditional kernels
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discriminant analysis on Embedded Manifold
European Conference on Computer Vision, 2004Co-Authors: Shuicheng Yan, Hongjiang Zhang, Benyu Zhang, Qiansheng ChengAbstract:Previous Manifold learning algorithms mainly focus on uncovering the low dimensional geometry structure from a set of samples that lie on or nearly on a Manifold in an unsupervised manner. However, the representations from unsupervised learning are not always optimal in discriminating capability. In this paper, a novel algorithm is introduced to conduct discriminant analysis in term of the Embedded Manifold structure. We propose a novel clustering algorithm, called Intra-Cluster Balanced K-Means (ICBKM), which ensures that there are balanced samples for the classes in a cluster; and the local discriminative features for all clusters are simultaneously calculated by following the global Fisher criterion. Compared to the traditional linear/kernel discriminant analysis algorithms, ours has the following characteristics: 1) it is approximately a locally linear yet globally nonlinear discriminant analyzer; 2) it can be considered a special Kernel-DA with geometry-adaptive-kernel, in contrast to traditional KDA whose kernel is independent to the samples; and 3) its computation and memory cost are reduced a great deal compared to traditional KDA, especially for the cases with large number of samples. It does not need to store the original samples for computing the low dimensional representation for new data. The evaluation on toy problem shows that it is effective in deriving discriminative representations for the problem with nonlinear classification hyperplane. When applied to the face recognition problem, it is shown that, compared with LDA and traditional KDA on YALE and PIE databases, the proposed algorithm significantly outperforms LDA and
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ECCV (1) - Discriminant Analysis on Embedded Manifold
Lecture Notes in Computer Science, 2004Co-Authors: Shuicheng Yan, Hongjiang Zhang, Benyu Zhang, Qiansheng ChengAbstract:Previous Manifold learning algorithms mainly focus on uncovering the low dimensional geometry structure from a set of samples that lie on or nearly on a Manifold in an unsupervised manner. However, the representations from unsupervised learning are not always optimal in discriminating capability. In this paper, a novel algorithm is introduced to conduct discriminant analysis in term of the Embedded Manifold structure. We propose a novel clustering algorithm, called Intra-Cluster Balanced K-Means (ICBKM), which ensures that there are balanced samples for the classes in a cluster; and the local discriminative features for all clusters are simultaneously calculated by following the global Fisher criterion. Compared to the traditional linear/kernel discriminant analysis algorithms, ours has the following characteristics: 1) it is approximately a locally linear yet globally nonlinear discriminant analyzer; 2) it can be considered a special Kernel-DA with geometry-adaptive-kernel, in contrast to traditional KDA whose kernel is independent to the samples; and 3) its computation and memory cost are reduced a great deal compared to traditional KDA, especially for the cases with large number of samples. It does not need to store the original samples for computing the low dimensional representation for new data. The evaluation on toy problem shows that it is effective in deriving discriminative representations for the problem with nonlinear classification hyperplane. When applied to the face recognition problem, it is shown that, compared with LDA and traditional KDA on YALE and PIE databases, the proposed algorithm significantly outperforms LDA and