1,218 research outputs found
Spectral Convergence of the connection Laplacian from random samples
Spectral methods that are based on eigenvectors and eigenvalues of discrete
graph Laplacians, such as Diffusion Maps and Laplacian Eigenmaps are often used
for manifold learning and non-linear dimensionality reduction. It was
previously shown by Belkin and Niyogi \cite{belkin_niyogi:2007} that the
eigenvectors and eigenvalues of the graph Laplacian converge to the
eigenfunctions and eigenvalues of the Laplace-Beltrami operator of the manifold
in the limit of infinitely many data points sampled independently from the
uniform distribution over the manifold. Recently, we introduced Vector
Diffusion Maps and showed that the connection Laplacian of the tangent bundle
of the manifold can be approximated from random samples. In this paper, we
present a unified framework for approximating other connection Laplacians over
the manifold by considering its principle bundle structure. We prove that the
eigenvectors and eigenvalues of these Laplacians converge in the limit of
infinitely many independent random samples. We generalize the spectral
convergence results to the case where the data points are sampled from a
non-uniform distribution, and for manifolds with and without boundary
An Invitation to Higher Gauge Theory
In this easy introduction to higher gauge theory, we describe parallel
transport for particles and strings in terms of 2-connections on 2-bundles.
Just as ordinary gauge theory involves a gauge group, this generalization
involves a gauge '2-group'. We focus on 6 examples. First, every abelian Lie
group gives a Lie 2-group; the case of U(1) yields the theory of U(1) gerbes,
which play an important role in string theory and multisymplectic geometry.
Second, every group representation gives a Lie 2-group; the representation of
the Lorentz group on 4d Minkowski spacetime gives the Poincar\'e 2-group, which
leads to a spin foam model for Minkowski spacetime. Third, taking the adjoint
representation of any Lie group on its own Lie algebra gives a 'tangent
2-group', which serves as a gauge 2-group in 4d BF theory, which has
topological gravity as a special case. Fourth, every Lie group has an 'inner
automorphism 2-group', which serves as the gauge group in 4d BF theory with
cosmological constant term. Fifth, every Lie group has an 'automorphism
2-group', which plays an important role in the theory of nonabelian gerbes. And
sixth, every compact simple Lie group gives a 'string 2-group'. We also touch
upon higher structures such as the 'gravity 3-group' and the Lie 3-superalgebra
that governs 11-dimensional supergravity.Comment: 60 pages, based on lectures at the 2nd School and Workshop on Quantum
Gravity and Quantum Geometry at the 2009 Corfu Summer Institut
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