7,391 research outputs found
A global approach to the refinement of manifold data
A refinement of manifold data is a computational process, which produces a
denser set of discrete data from a given one. Such refinements are closely
related to multiresolution representations of manifold data by pyramid
transforms, and approximation of manifold-valued functions by repeated
refinements schemes. Most refinement methods compute each refined element
separately, independently of the computations of the other elements. Here we
propose a global method which computes all the refined elements simultaneously,
using geodesic averages. We analyse repeated refinements schemes based on this
global approach, and derive conditions guaranteeing strong convergence.Comment: arXiv admin note: text overlap with arXiv:1407.836
Definability and stability of multiscale decompositions for manifold-valued data
We discuss multiscale representations of discrete manifold-valued data. As it
turns out that we cannot expect general manifold-analogues of biorthogonal
wavelets to possess perfect reconstruction, we focus our attention on those
constructions which are based on upscaling operators which are either
interpolating or midpoint-interpolating. For definable multiscale
decompositions we obtain a stability result
emgr - The Empirical Gramian Framework
System Gramian matrices are a well-known encoding for properties of
input-output systems such as controllability, observability or minimality.
These so-called system Gramians were developed in linear system theory for
applications such as model order reduction of control systems. Empirical
Gramian are an extension to the system Gramians for parametric and nonlinear
systems as well as a data-driven method of computation. The empirical Gramian
framework - emgr - implements the empirical Gramians in a uniform and
configurable manner, with applications such as Gramian-based (nonlinear) model
reduction, decentralized control, sensitivity analysis, parameter
identification and combined state and parameter reduction
Smoothness of Nonlinear and Non-Separable Subdivision Schemes
We study in this paper nonlinear subdivision schemes in a multivariate
setting allowing arbitrary dilation matrix. We investigate the convergence of
such iterative process to some limit function. Our analysis is based on some
conditions on the contractivity of the associated scheme for the differences.
In particular, we show the regularity of the limit function, in and
Sobolev spaces
H\"older Regularity of Geometric Subdivision Schemes
We present a framework for analyzing non-linear -valued
subdivision schemes which are geometric in the sense that they commute with
similarities in . It admits to establish
-regularity for arbitrary schemes of this type, and
-regularity for an important subset thereof, which includes all
real-valued schemes. Our results are constructive in the sense that they can be
verified explicitly for any scheme and any given set of initial data by a
universal procedure. This procedure can be executed automatically and
rigorously by a computer when using interval arithmetics.Comment: 31 pages, 1 figur
Multiscale Representations for Manifold-Valued Data
We describe multiscale representations for data observed on equispaced grids and taking values in manifolds such as the sphere , the special orthogonal group , the positive definite matrices , and the Grassmann manifolds . The representations are based on the deployment of Deslauriers--Dubuc and average-interpolating pyramids "in the tangent plane" of such manifolds, using the and maps of those manifolds. The representations provide "wavelet coefficients" which can be thresholded, quantized, and scaled in much the same way as traditional wavelet coefficients. Tasks such as compression, noise removal, contrast enhancement, and stochastic simulation are facilitated by this representation. The approach applies to general manifolds but is particularly suited to the manifolds we consider, i.e., Riemannian symmetric spaces, such as , , , where the and maps are effectively computable. Applications to manifold-valued data sources of a geometric nature (motion, orientation, diffusion) seem particularly immediate. A software toolbox, SymmLab, can reproduce the results discussed in this paper
Strong and weak order in averaging for SPDEs
We show an averaging result for a system of stochastic evolution equations of
parabolic type with slow and fast time scales. We derive explicit bounds for
the approximation error with respect to the small parameter defining the fast
time scale. We prove that the slow component of the solution of the system
converges towards the solution of the averaged equation with an order of
convergence is 1/2 in a strong sense - approximation of trajectories - and 1 in
a weak sense - approximation of laws. These orders turn out to be the same as
for the SDE case
An Online Unsupervised Structural Plasticity Algorithm for Spiking Neural Networks
In this article, we propose a novel Winner-Take-All (WTA) architecture
employing neurons with nonlinear dendrites and an online unsupervised
structural plasticity rule for training it. Further, to aid hardware
implementations, our network employs only binary synapses. The proposed
learning rule is inspired by spike time dependent plasticity (STDP) but differs
for each dendrite based on its activation level. It trains the WTA network
through formation and elimination of connections between inputs and synapses.
To demonstrate the performance of the proposed network and learning rule, we
employ it to solve two, four and six class classification of random Poisson
spike time inputs. The results indicate that by proper tuning of the inhibitory
time constant of the WTA, a trade-off between specificity and sensitivity of
the network can be achieved. We use the inhibitory time constant to set the
number of subpatterns per pattern we want to detect. We show that while the
percentage of successful trials are 92%, 88% and 82% for two, four and six
class classification when no pattern subdivisions are made, it increases to
100% when each pattern is subdivided into 5 or 10 subpatterns. However, the
former scenario of no pattern subdivision is more jitter resilient than the
later ones.Comment: 11 pages, 10 figures, journa
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