6,738 research outputs found
Dissipative numerical schemes on Riemannian manifolds with applications to gradient flows
This paper concerns an extension of discrete gradient methods to
finite-dimensional Riemannian manifolds termed discrete Riemannian gradients,
and their application to dissipative ordinary differential equations. This
includes Riemannian gradient flow systems which occur naturally in optimization
problems. The Itoh--Abe discrete gradient is formulated and applied to gradient
systems, yielding a derivative-free optimization algorithm. The algorithm is
tested on two eigenvalue problems and two problems from manifold valued
imaging: InSAR denoising and DTI denoising.Comment: Post-revision version. To appear in SIAM Journal on Scientific
Computin
Locality Preserving Projections for Grassmann manifold
Learning on Grassmann manifold has become popular in many computer vision
tasks, with the strong capability to extract discriminative information for
imagesets and videos. However, such learning algorithms particularly on
high-dimensional Grassmann manifold always involve with significantly high
computational cost, which seriously limits the applicability of learning on
Grassmann manifold in more wide areas. In this research, we propose an
unsupervised dimensionality reduction algorithm on Grassmann manifold based on
the Locality Preserving Projections (LPP) criterion. LPP is a commonly used
dimensionality reduction algorithm for vector-valued data, aiming to preserve
local structure of data in the dimension-reduced space. The strategy is to
construct a mapping from higher dimensional Grassmann manifold into the one in
a relative low-dimensional with more discriminative capability. The proposed
method can be optimized as a basic eigenvalue problem. The performance of our
proposed method is assessed on several classification and clustering tasks and
the experimental results show its clear advantages over other Grassmann based
algorithms.Comment: Accepted by IJCAI 201
Nondegeneracy of the Ground State for Nonrelativistic Lee Model
In the present work, we first briefly sketch construction of the
nonrelativistic Lee model on Riemannian manifolds, introduced in our previous
works. In this approach, the renormalized resolvent of the system is expressed
in terms of a well-defined operator, called the principal operator, so as to
obtain a finite formulation. Then, we show that the ground state of the
nonrelativistic Lee model on a compact Riemannian manifolds is nondegenerate
using the explicit expression of the principal operator that we obtained. This
is achieved by combining heat kernel methods with positivity improving
semi-group approach and then applying these tools directly to the principal
operator, rather than the Hamiltonian, without using cut-offs.Comment: 16 pages, typos are corrected, abstract and some sentences in the
text are improved. Appears in Journal of Mathematical Physics, volume 55,
issue 8 (2014
Existence of periodic orbits for geodesible vector fields on closed 3-manifolds
In this paper we deal with the existence of periodic orbits of geodesible
vector fields on closed 3-manifolds. A vector field is geodesible if there
exists a Riemannian metric on the ambient manifold making its orbits geodesics.
In particular, Reeb vector fields and vector fields that admit a global section
are geodesible. We will classify the closed 3-manifolds that admit aperiodic
volume preserving real analytic geodesible vector fields, and prove the
existence of periodic orbits for real analytic geodesible vector fields (not
volume preserving), when the 3-manifold is not a torus bundle over the circle.
We will also prove the existence of periodic orbits of C2 geodesible vector
fields in some closed 3-manifolds
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