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56
Consistency of the group lasso and multiple kernel learning
 JOURNAL OF MACHINE LEARNING RESEARCH
, 2007
"... We consider the leastsquare regression problem with regularization by a block 1norm, i.e., a sum of Euclidean norms over spaces of dimensions larger than one. This problem, referred to as the group Lasso, extends the usual regularization by the 1norm where all spaces have dimension one, where it ..."
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Cited by 158 (26 self)
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We consider the leastsquare regression problem with regularization by a block 1norm, i.e., a sum of Euclidean norms over spaces of dimensions larger than one. This problem, referred to as the group Lasso, extends the usual regularization by the 1norm where all spaces have dimension one, where it is commonly referred to as the Lasso. In this paper, we study the asymptotic model consistency of the group Lasso. We derive necessary and sufficient conditions for the consistency of group Lasso under practical assumptions, such as model misspecification. When the linear predictors and Euclidean norms are replaced by functions and reproducing kernel Hilbert norms, the problem is usually referred to as multiple kernel learning and is commonly used for learning from heterogeneous data sources and for non linear variable selection. Using tools from functional analysis, and in particular covariance operators, we extend the consistency results to this infinite dimensional case and also propose an adaptive scheme to obtain a consistent model estimate, even when the necessary condition required for the non adaptive scheme is not satisfied.
CoherenceEnhancing Diffusion Filtering
, 1999
"... The completion of interrupted lines or the enhancement of flowlike structures is a challenging task in computer vision, human vision, and image processing. We address this problem by presenting a multiscale method in which a nonlinear diffusion filter is steered by the socalled interest operato ..."
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Cited by 87 (2 self)
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The completion of interrupted lines or the enhancement of flowlike structures is a challenging task in computer vision, human vision, and image processing. We address this problem by presenting a multiscale method in which a nonlinear diffusion filter is steered by the socalled interest operator (secondmoment matrix, structure tensor). An mdimensional formulation of this method is analysed with respect to its wellposedness and scalespace properties. An efficient scheme is presented which uses a stabilization by a semiimplicit additive operator splitting (AOS), and the scalespace behaviour of this method is illustrated by applying it to both 2D and 3D images.
Convergence to Equilibrium for the Relaxation Approximations of Conservation Laws
, 1996
"... We study the Cauchy problem for 2\Theta2 semilinear and quasilinear hyperbolic systems with a singular relaxation term. Special comparison and compactness properties are established by assuming the subcharacteristic condition. Therefore we can prove the convergence to equilibrium of the solutions of ..."
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Cited by 72 (13 self)
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We study the Cauchy problem for 2\Theta2 semilinear and quasilinear hyperbolic systems with a singular relaxation term. Special comparison and compactness properties are established by assuming the subcharacteristic condition. Therefore we can prove the convergence to equilibrium of the solutions of these problems as the singular perturbation parameter tends to zero. This research was strongly motivated by the recent numerical investigations of S. Jin and Z. Xin on the relaxation schemes for conservation laws. 1. Introduction In this paper we are interested to the relaxation behaviour of the following system of hyperbolic conservation laws with a singular perturbation source (1.1) ae @ t u + @ x v = 0 ; @ t v + @ x oe(u) = \Gamma 1 " (v \Gamma f(u)) (" ? 0); for (x; t) 2 IR \Theta (0; 1). Here oe, f are some given smooth functions such that oe 0 (u) ( ? 0), f(0) = 0. The system (1.1) is equivalent to the onedimensional perturbed wave equation (1.2) @ tt w \Gamma @ x oe(@ x...
Theoretical Foundations Of Anisotropic Diffusion In Image Processing
 Computing, Suppl
, 1996
"... A frequent problem in lowlevel vision consists of eliminating noise and smallscale details from an image while still preserving or even enhancing the edge structure. Nonlinear anisotropic diffusion filtering may be one possibility to achieve these goals. The objective of the present paper is to ..."
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Cited by 44 (13 self)
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A frequent problem in lowlevel vision consists of eliminating noise and smallscale details from an image while still preserving or even enhancing the edge structure. Nonlinear anisotropic diffusion filtering may be one possibility to achieve these goals. The objective of the present paper is to review the author's results on a scalespace interpretation of a class of diffusion filters which comprises also several nonlinear anisotropic models. It is demonstrated that these models  which use an adapted diffusion tensor instead of a scalar diffusivity  offer advantages over isotropic filters. Most of the restoration and scalespace properties carry over from the continuous to the discrete case. Applications are presented ranging from preprocessing of medical images and postprocessing of fluctuating numerical data to visualizing quality relevant features for the grading of wood surfaces and fabrics.
ScaleSpace Properties of Nonlinear Diffusion Filtering with a Diffusion Tensor
 Laboratory of Technomathematics, University of Kaiserslautern, P.O
, 1994
"... In spite of its lack of theoretical justification, nonlinear diffusion filtering has become a powerful image enhancement tool in recent years. The goal of the present paper is to provide a mathematical foundation for continuous nonlinear diffusion filtering as a scalespace transformation which is f ..."
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Cited by 20 (3 self)
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In spite of its lack of theoretical justification, nonlinear diffusion filtering has become a powerful image enhancement tool in recent years. The goal of the present paper is to provide a mathematical foundation for continuous nonlinear diffusion filtering as a scalespace transformation which is flexible enough to simplify images without loosing the capability of enhancing edges. By studying the Lyapunov functionals, it is shown that nonlinear diffusion reduces L p norms and central moments and increases the entropy of images. The proposed anisotropic class utilizes a diffusion tensor which may be adapted to the image structure. It permits existence, uniqueness and regularity results, the solution depends continuously on the initial image, and it satisfies an extremum principle. All considerations include linear and certain nonlinear isotropic models and apply to m dimensional vectorvalued images. The results are juxtaposed to linear and morphological scalespaces. . Keywords....
HighDimensional NonLinear Variable Selection through Hierarchical Kernel Learning
, 2009
"... We consider the problem of highdimensional nonlinear variable selection for supervised learning. Our approach is based on performing linear selection among exponentially many appropriately defined positive definite kernels that characterize nonlinear interactions between the original variables. T ..."
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Cited by 18 (5 self)
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We consider the problem of highdimensional nonlinear variable selection for supervised learning. Our approach is based on performing linear selection among exponentially many appropriately defined positive definite kernels that characterize nonlinear interactions between the original variables. To select efficiently from these many kernels, we use the natural hierarchical structure of the problem to extend the multiple kernel learning framework to kernels that can be embedded in a directed acyclic graph; we show that it is then possible to perform kernel selection through a graphadapted sparsityinducing norm, in polynomial time in the number of selected kernels. Moreover, we study the consistency of variable selection in highdimensional settings, showing that under certain assumptions, our regularization framework allows a number of irrelevant variables which is exponential in the number of observations. Our simulations on synthetic datasets and datasets from the UCI repository show stateoftheart predictive performance for nonlinear regression problems. 1
A Generalized Index Theorem for Morse–Sturm Systems and Applications to semiRiemannian Geometry
 Asian Journal of Mathematics
"... ABSTRACT. We prove an extension of the Index Theorem for Morse–Sturm systems of the form −V ′ ′ + RV = 0, where R is symmetric with respect to a (non positive) symmetric bilinear form, and thus the corresponding differential operator is not selfadjoint. The result is then applied to the case of a J ..."
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Cited by 18 (13 self)
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ABSTRACT. We prove an extension of the Index Theorem for Morse–Sturm systems of the form −V ′ ′ + RV = 0, where R is symmetric with respect to a (non positive) symmetric bilinear form, and thus the corresponding differential operator is not selfadjoint. The result is then applied to the case of a Jacobi equation along a geodesic in a Lorentzian manifold, obtaining an extension of the Morse Index Theorem for Lorentzian geodesics with variable initial endpoints. Given a Lorentzian manifold (M, g), we consider a geodesic γ in M starting orthogonally to a smooth submanifold P of M. Under suitable hypotheses, satisfied, for instance, if (M, g) is stationary, the theorem gives an equality between the index of the second variation of the action functional f at γ and the sum of the Maslov index of γ with the index of the metric g on P. Under generic circumstances, the Maslov index of γ is given by an algebraic count of the Pfocal points along γ. Using the Maslov index, we obtain the global Morse relations for geodesics between two fixed points in a stationary Lorentzian manifold. 1.
THE MORSE INDEX THEOREM IN SEMIRIEMANNIAN GEOMETRY
, 2000
"... ABSTRACT. We prove a semiRiemannian version of the celebrated Morse Index Theorem for geodesics in semiRiemannian manifolds; we consider the general case of both endpoints variable on two submanifolds. The key role of the theory is played by the notion of the Maslov index of a semiRiemannian geod ..."
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Cited by 13 (7 self)
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ABSTRACT. We prove a semiRiemannian version of the celebrated Morse Index Theorem for geodesics in semiRiemannian manifolds; we consider the general case of both endpoints variable on two submanifolds. The key role of the theory is played by the notion of the Maslov index of a semiRiemannian geodesic, which is a homological invariant and it substitutes the notion of geometric index in Riemannian geometry. Under generic circumstances, the Maslov index of a geodesic is computed as a sort of algebraic count of the conjugate points along the geodesic. For non positive definite metrics the index of the index form is always infinite; in this paper we prove that the space of all variations of a given geodesic has a natural splitting into two infinite dimensional subspaces, and the Maslov index is given by the difference of the index and the coindex of the restriction of the index form to these subspaces. In the case of variable endpoints, two suitable correction terms, defined in terms of the endmanifolds, are added to the equality. Using appropriate change of variables, the theory is entirely extended to the more general case of symplectic differential systems, that can be obtained as linearizations of the Hamilton equations. The main results proven in this paper were announced in [23].
Extremal Eigenvalue Problems For Composite Membranes, I
, 1990
"... : Given an open bounded connected set\Omega ae R N and a prescribed amount of two homogeneous materials of different density, for small k we characterize the distribution of the two materials in\Omega that extremizes the kth eigenvalue of the resulting clamped membrane. We show that these extrem ..."
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Cited by 13 (3 self)
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: Given an open bounded connected set\Omega ae R N and a prescribed amount of two homogeneous materials of different density, for small k we characterize the distribution of the two materials in\Omega that extremizes the kth eigenvalue of the resulting clamped membrane. We show that these extremizers vary continuously with the proportion of the two constituents. The characterization of the extremizers in terms of level sets of associated eigenfunctions provides geometric information on their respective interfaces. Each of these results generalizes to N dimensions the now classical one dimensional work of M.G. Krein. 1 1. INTRODUCTION Within the class of fixed endpoint strings on the interval (0; 1) with density between ff and fi and mass equal to fffl + fi(1 \Gamma fl), M.G. Krein [19] was able to isolate those with the largest or smallest kth natural frequency. More precisely, denoting by k (ae) the kth Dirichlet eigenvalue of the string with density ae, and by ae fl k (ae ...