338 research outputs found
Extrinsic Methods for Coding and Dictionary Learning on Grassmann Manifolds
Sparsity-based representations have recently led to notable results in
various visual recognition tasks. In a separate line of research, Riemannian
manifolds have been shown useful for dealing with features and models that do
not lie in Euclidean spaces. With the aim of building a bridge between the two
realms, we address the problem of sparse coding and dictionary learning over
the space of linear subspaces, which form Riemannian structures known as
Grassmann manifolds. To this end, we propose to embed Grassmann manifolds into
the space of symmetric matrices by an isometric mapping. This in turn enables
us to extend two sparse coding schemes to Grassmann manifolds. Furthermore, we
propose closed-form solutions for learning a Grassmann dictionary, atom by
atom. Lastly, to handle non-linearity in data, we extend the proposed Grassmann
sparse coding and dictionary learning algorithms through embedding into Hilbert
spaces.
Experiments on several classification tasks (gender recognition, gesture
classification, scene analysis, face recognition, action recognition and
dynamic texture classification) show that the proposed approaches achieve
considerable improvements in discrimination accuracy, in comparison to
state-of-the-art methods such as kernelized Affine Hull Method and
graph-embedding Grassmann discriminant analysis.Comment: Appearing in International Journal of Computer Visio
A Framework for Non-Gaussian Functional Integrals with Applications
Functional integrals can be defined in terms of families of locally compact
topological groups and their associated Banach-valued Haar integrals. The
definition forgoes the goal of constructing a genuine measure on the domain of
integration, but instead provides for a topological realization of localization
in the infinite-dimensional domain. This leads to measurable subspaces that
characterize meaningful functional integrals and a scheme that possesses
significant potential for constructing and representing non-commutative Banach
algebras.
The framework includes, within a broader structure, other successful
approaches to define functional integrals in restricted cases, and it suggests
new and potentially useful functional integrals that go beyond the standard
Gaussian case. In particular, functional integrals based on skew-Hermitian and
K\"{a}hler quadratic forms are defined and developed. Also defined are
gamma-type and Poisson-type functional integrals based on linear forms
suggested by the gamma probability distribution. These and their
generalizations are expected to play a leading role in generating
-algebras of quantum systems. Several applications and implications are
explored.Comment: This is the first of two papers representing an expanded version of
arXiv:1308.106
Massive Nonlinear Sigma Models and BPS Domain Walls in Harmonic Superspace
Four-dimensional massive N=2 nonlinear sigma models and BPS wall solutions
are studied in the off-shell harmonic superspace approach in which N=2
supersymmetry is manifest. The general nonlinear sigma model can be described
by an analytic harmonic potential which is the hyper-Kahler analog of the
Kahler potential in N=1 theory. We examine the massive nonlinear sigma model
with multi-center four-dimensional target hyper-Kahler metrics and derive the
corresponding BPS equation. We study in some detail two particular cases with
the Taub-NUT and double Taub-NUT metrics. The latter embodies, as its two
separate limits, both Taub-NUT and Eguchi-Hanson metrics. We find that domain
wall solutions exist only in the double Taub-NUT case including its
Eguchi-Hanson limit.Comment: 35 pages, 4 figures, minor corrections and references added, to
appear in NP
Paraconformal geometry of th order ODEs, and exotic holonomy in dimension four
We characterise th order ODEs for which the space of solutions is
equipped with a particular paraconformal structure in the sense of \cite{BE},
that is a splitting of the tangent bundle as a symmetric tensor product of
rank-two vector bundles. This leads to the vanishing of quantities
constructed from of the ODE.
If the paraconformal structure is shown to be equivalent to the exotic
holonomy of Bryant. If , or and admits a
torsion--free connection compatible with the paraconformal structure then the
ODE is trivialisable by point or contact transformations respectively.
If or 3 admits an affine paraconformal connection with no torsion.
In these cases additional constraints can be imposed on the ODE so that
admits a projective structure if , or an Einstein--Weyl structure if
. The third order ODE can in this case be reconstructed from the
Einstein--Weyl data.Comment: Theorem 1.2 strengthened and its proof clarified. Theorem 1.3
generalised to all dimensions, updated references, an example of 5th order
ODE on the space of conics in added, connection with
Doubrov-Wilczynski invariants clarified. Final version, to appear in Journal
of Geometry and Physic
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