83,842 research outputs found
On the optimality of shape and data representation in the spectral domain
A proof of the optimality of the eigenfunctions of the Laplace-Beltrami
operator (LBO) in representing smooth functions on surfaces is provided and
adapted to the field of applied shape and data analysis. It is based on the
Courant-Fischer min-max principle adapted to our case. % The theorem we present
supports the new trend in geometry processing of treating geometric structures
by using their projection onto the leading eigenfunctions of the decomposition
of the LBO. Utilisation of this result can be used for constructing numerically
efficient algorithms to process shapes in their spectrum. We review a couple of
applications as possible practical usage cases of the proposed optimality
criteria. % We refer to a scale invariant metric, which is also invariant to
bending of the manifold. This novel pseudo-metric allows constructing an LBO by
which a scale invariant eigenspace on the surface is defined. We demonstrate
the efficiency of an intermediate metric, defined as an interpolation between
the scale invariant and the regular one, in representing geometric structures
while capturing both coarse and fine details. Next, we review a numerical
acceleration technique for classical scaling, a member of a family of
flattening methods known as multidimensional scaling (MDS). There, the
optimality is exploited to efficiently approximate all geodesic distances
between pairs of points on a given surface, and thereby match and compare
between almost isometric surfaces. Finally, we revisit the classical principal
component analysis (PCA) definition by coupling its variational form with a
Dirichlet energy on the data manifold. By pairing the PCA with the LBO we can
handle cases that go beyond the scope defined by the observation set that is
handled by regular PCA
Explicit constructions of operator scaling Gaussian fields
We propose an explicit way to generate a large class of Operator scaling
Gaussian random fields (OSGRF). Such fields are anisotropic generalizations of
selfsimilar fields. More specifically, we are able to construct any Gaussian
field belonging to this class with given Hurst index and exponent. Our
construction provides - for simulations of texture as well as for detection of
anisotropies in an image - a large class of models with controlled anisotropic
geometries and structures
On the Scaling Limits of Determinantal Point Processes with Kernels Induced by Sturm-Liouville Operators
By applying an idea of Borodin and Olshanski [J. Algebra 313 (2007), 40-60],
we study various scaling limits of determinantal point processes with trace
class projection kernels given by spectral projections of selfadjoint
Sturm-Liouville operators. Instead of studying the convergence of the kernels
as functions, the method directly addresses the strong convergence of the
induced integral operators. We show that, for this notion of convergence, the
Dyson, Airy, and Bessel kernels are universal in the bulk, soft-edge, and
hard-edge scaling limits. This result allows us to give a short and unified
derivation of the known formulae for the scaling limits of the classical random
matrix ensembles with unitary invariance, that is, the Gaussian unitary
ensemble (GUE), the Wishart or Laguerre unitary ensemble (LUE), and the MANOVA
(multivariate analysis of variance) or Jacobi unitary ensemble (JUE)
Microscopic Spectrum of the Wilson Dirac Operator
We calculate the leading contribution to the spectral density of the Wilson
Dirac operator using chiral perturbation theory where volume and lattice
spacing corrections are given by universal scaling functions. We find
analytical expressions for the spectral density on the scale of the average
level spacing, and introduce a chiral Random Matrix Theory that reproduces
these results. Our work opens up a novel approach to the infinite volume limit
of lattice gauge theory at finite lattice spacing and new ways to extract
coefficients of Wilson chiral perturbation theory.Comment: 4 pages, 3 figures, refs added, version to appear in Phys. Rev. Let
A Spectral Theory for Tensors
In this paper we propose a general spectral theory for tensors. Our proposed
factorization decomposes a tensor into a product of orthogonal and scaling
tensors. At the same time, our factorization yields an expansion of a tensor as
a summation of outer products of lower order tensors . Our proposed
factorization shows the relationship between the eigen-objects and the
generalised characteristic polynomials. Our framework is based on a consistent
multilinear algebra which explains how to generalise the notion of matrix
hermicity, matrix transpose, and most importantly the notion of orthogonality.
Our proposed factorization for a tensor in terms of lower order tensors can be
recursively applied so as to naturally induces a spectral hierarchy for
tensors.Comment: The paper is an updated version of an earlier versio
Semiclassical Prediction of Large Spectral Fluctuations in Interacting Kicked Spin Chains
While plenty of results have been obtained for single-particle quantum
systems with chaotic dynamics through a semiclassical theory, much less is
known about quantum chaos in the many-body setting. We contribute to recent
efforts to make a semiclassical analysis of many-body systems feasible. This is
nontrivial due to both the enormous density of states and the exponential
proliferation of periodic orbits with the number of particles. As a model
system we study kicked interacting spin chains employing semiclassical methods
supplemented by a newly developed duality approach. We show that for this model
the line between integrability and chaos becomes blurred. Due to the
interaction structure the system features (non-isolated) manifolds of periodic
orbits possessing highly correlated, collective dynamics. As with the invariant
tori in integrable systems, their presence lead to significantly enhanced
spectral fluctuations, which by order of magnitude lie in-between integrable
and chaotic cases.Comment: 42 pages, 19 figure
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