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95
Diffusion Kernels on Statistical Manifolds
, 2004
"... A family of kernels for statistical learning is introduced that exploits the geometric structure of statistical models. The kernels are based on the heat equation on the Riemannian manifold defined by the Fisher information metric associated with a statistical family, and generalize the Gaussian ker ..."
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Cited by 92 (6 self)
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A family of kernels for statistical learning is introduced that exploits the geometric structure of statistical models. The kernels are based on the heat equation on the Riemannian manifold defined by the Fisher information metric associated with a statistical family, and generalize the Gaussian kernel of Euclidean space. As an important special case, kernels based on the geometry of multinomial families are derived, leading to kernelbased learning algorithms that apply naturally to discrete data. Bounds on covering numbers and Rademacher averages for the kernels are proved using bounds on the eigenvalues of the Laplacian on Riemannian manifolds. Experimental results are presented for document classification, for which the use of multinomial geometry is natural and well motivated, and improvements are obtained over the standard use of Gaussian or linear kernels, which have been the standard for text classification.
Twisted connected sums and special Riemannian holonomy
 J. Reine Angew. Math
"... Abstract. We give a new, connected sum construction of Riemannian metrics with special holonomy G2 on compact 7manifolds. The construction is based on a gluing theorem for appropriate elliptic partial differential equations. As a prerequisite, we also obtain asymptotically cylindrical Riemannian ma ..."
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Cited by 47 (5 self)
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Abstract. We give a new, connected sum construction of Riemannian metrics with special holonomy G2 on compact 7manifolds. The construction is based on a gluing theorem for appropriate elliptic partial differential equations. As a prerequisite, we also obtain asymptotically cylindrical Riemannian manifolds with holonomy SU(3) building up on the work of Tian and Yau. Examples of new topological types of compact 7manifolds with holonomy G2 are constructed using Fano 3folds. The purpose of this paper is to give a new construction of compact 7dimensional Riemannian manifolds with holonomy group G2. The holonomy group of a Riemannian manifold is the group of isometries of a tangent space generated by parallel transport using the Levi–Civita connection over closed paths based at a point. For an oriented ndimensional manifold the holonomy group may be identified as a subgroup of SO(n). If there is a structure on a manifold defined by a tensor field and parallel with respect to the Levi–Civita connection then the holonomy may be a proper subgroup of SO(n) (it is just the subgroup leaving invariant the corresponding tensor on R n). There is essentially just one possibility for such holonomy reduction in odd dimensions, as follows from the wellknown Berger
LaplaceSpectra as Fingerprints for Shape Matching
, 2005
"... This paper introduces a method to extract fingerprints of any surface or solid object by taking the eigenvalues of its respective LaplaceBeltrami operator. Using an object's spectrum (i.e. the family of its eigenvalues) as a fingerprint for its shape is motivated by the fact that the related ei ..."
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Cited by 33 (5 self)
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This paper introduces a method to extract fingerprints of any surface or solid object by taking the eigenvalues of its respective LaplaceBeltrami operator. Using an object's spectrum (i.e. the family of its eigenvalues) as a fingerprint for its shape is motivated by the fact that the related eigenvalues are isometry invariants of the object. Employing the LaplaceBeltrami spectra (not the spectra of the mesh Laplacian) as fingerprints of surfaces and solids is a novel approach in the field of geometric modeling and computer graphics. Those spectra can be calculated for any representation of the geometric object (e.g. NURBS or any parametrized or implicitly represented surface or even for polyhedra). Since the spectrum is an isometry invariant of the respective object this fingerprint is also independent of the spatial position. Additionally the eigenvalues can be normalized so that scaling factors for the geometric object can be obtained easily. Therefore checking if two objects are isometric needs no prior alignment (registration/localization) of the objects, but only a comparison of their spectra. With the help of such fingerprints it is possible to support copyright protection, database retrieval and quality assessment of digital data representing surfaces and solids.
Harmonic analysis and propagators on homogeneous spaces
 Phys. Rep
, 1990
"... 2. The heat kernel and the Schwinger—DeWitt expansion 7 Riemann—Liouville integral 48 3. Example: the Einstein universe II 8.3. The heat kernel Ofl S~in terms of fractional deriva4. The eigenfunction expansion on a homogeneous space 14 tives 5)) ..."
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Cited by 24 (2 self)
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2. The heat kernel and the Schwinger—DeWitt expansion 7 Riemann—Liouville integral 48 3. Example: the Einstein universe II 8.3. The heat kernel Ofl S~in terms of fractional deriva4. The eigenfunction expansion on a homogeneous space 14 tives 5))
Birkhoff normal form and Hamiltonian PDEs, Partial differential equations and applications
 Sémin. Congr
, 2007
"... Abstract. — These notes are based on lectures held at the Lanzhou university (China) during a CIMPA summer school in july 2004 but benefit from recent devellopements. Our aim is to explain some normal form technics that allow to study the long time behaviour of the solutions of Hamiltonian perturbat ..."
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Cited by 18 (9 self)
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Abstract. — These notes are based on lectures held at the Lanzhou university (China) during a CIMPA summer school in july 2004 but benefit from recent devellopements. Our aim is to explain some normal form technics that allow to study the long time behaviour of the solutions of Hamiltonian perturbations of integrable systems. We are in particular interested with stability results. Our approach is centered on the Birkhoff normal form theorem that we first proved in finite dimension. Then, after giving some exemples of Hamiltonian PDEs, we present an abstract Birkhoff normal form theorem in infinite dimension and discuss the dynamical consequences for Hamiltonian PDEs. Résumé (Forme normale de Birkhoff et EDP Hamiltoniènnes) Ces notes sont basées sur un cours donné à l’université de Lanzhou (Chine) durant le mois de juillet 2004 dans le cadre d’une école d’été organisée par le CIMPA. Cette rédaction bénéficie aussi de développements plus récents. Le but est d’expliquer certaines techniques de forme normale qui permettent d’étudier le comportement pour des temps longs des solutions de perturbations Hamiltoniènnes de systèmes intégrables. Nous sommes en particulier intéressés par des résultats de stabilité. Notre approche est centrée sur le théorème de forme normale de Birkhoff que nous rappelons et démontrons d’abord en dimension finie. Ensuite, après avoir donné quelques exemples d’EDP Hamiltoniènnes, nous démontrons un théorème de forme normale de Birkhoff en dimension infinie et nous en discutons les applications à la dynamique des EDP Hamiltoniènnes.
Renormalisation and the BatalinVilkovisky formalism
"... ABSTRACT. This paper gives a way to renormalise certain quantum field theories on compact manifolds. Examples include YangMills theory (in dimension 4 only), ChernSimons theory and holomorphic ChernSimons theory. The method is within the framework of the BatalinVilkovisky formalism. ChernSimons ..."
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Cited by 16 (1 self)
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ABSTRACT. This paper gives a way to renormalise certain quantum field theories on compact manifolds. Examples include YangMills theory (in dimension 4 only), ChernSimons theory and holomorphic ChernSimons theory. The method is within the framework of the BatalinVilkovisky formalism. ChernSimons theory is renormalised in a way respecting all symmetries (up to homotopy). This yields an invariant of smooth manifolds: a certain algebraic structure on the cohomology of the manifold tensored with a Lie algebra, which is a “higher loop ” enrichment of the natural L∞ structure. 1.
Asymptotic expansion for the heat kernel for orbifolds
 Michigan Math. J
"... ABSTRACT. We study the relationship between the geometry and the Laplace spectrum of a Riemannian orbifold O via its heat kernel; as in the manifold case, the timezero asymptotic expansion of the heat kernel furnishes geometric information about O. In the case of a good Riemannian orbifold (i.e., a ..."
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Cited by 15 (3 self)
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ABSTRACT. We study the relationship between the geometry and the Laplace spectrum of a Riemannian orbifold O via its heat kernel; as in the manifold case, the timezero asymptotic expansion of the heat kernel furnishes geometric information about O. In the case of a good Riemannian orbifold (i.e., an orbifold arising as the orbit space of a manifold under the action of a discrete group of isometries), H. Donnelly [10] proved the existence of the heat kernel and constructed the asymptotic expansion for the heat trace. We extend Donnelly’s work to the case of general compact orbifolds. Moreover, in both the good case and the general case, we express the heat invariants in a form that clarifies the asymptotic contribution of each part of the singular set of the orbifold. We calculate several terms in the asymptotic expansion explicitly in the case of twodimensional orbifolds; we use these terms to prove that the spectrum distinguishes elements within various classes of twodimensional orbifolds.
Isospectral Manifolds with Different Local Geometries
, 2001
"... We construct several new classes of isospectral manifolds with different local geometries. After reviewing a theorem by Carolyn Gordon on isospectral torus bundles and presenting certain useful specialized versions (Chapter 1) we apply these tools to construct the first examples of isospectral fou ..."
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Cited by 15 (5 self)
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We construct several new classes of isospectral manifolds with different local geometries. After reviewing a theorem by Carolyn Gordon on isospectral torus bundles and presenting certain useful specialized versions (Chapter 1) we apply these tools to construct the first examples of isospectral fourdimensional manifolds which are not locally isometric (Chapter 2). Moreover, we construct the first examples of isospectral left invariant metrics on compact Lie groups (Chapter 3). Thereby we also obtain the first continuous isospectral families of globally homogeneous manifolds and the first examples of isospectral manifolds which are simply connected and irreducible. Finally, we construct the first pairs of isospectral manifolds which are conformally equivalent and not locally isometric (Chapter 4).
The compactification of a minimal submanifold in Euclidean space by the Gauss map, preprint
"... * Partially supported by an N.S.F. Postdoctoral Fellowship. 2 ..."
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Cited by 14 (0 self)
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* Partially supported by an N.S.F. Postdoctoral Fellowship. 2