27,534 research outputs found
A Riemannian Trust Region Method for the Canonical Tensor Rank Approximation Problem
The canonical tensor rank approximation problem (TAP) consists of
approximating a real-valued tensor by one of low canonical rank, which is a
challenging non-linear, non-convex, constrained optimization problem, where the
constraint set forms a non-smooth semi-algebraic set. We introduce a Riemannian
Gauss-Newton method with trust region for solving small-scale, dense TAPs. The
novelty of our approach is threefold. First, we parametrize the constraint set
as the Cartesian product of Segre manifolds, hereby formulating the TAP as a
Riemannian optimization problem, and we argue why this parametrization is among
the theoretically best possible. Second, an original ST-HOSVD-based retraction
operator is proposed. Third, we introduce a hot restart mechanism that
efficiently detects when the optimization process is tending to an
ill-conditioned tensor rank decomposition and which often yields a quick escape
path from such spurious decompositions. Numerical experiments show improvements
of up to three orders of magnitude in terms of the expected time to compute a
successful solution over existing state-of-the-art methods
Diversity Spectra of Spatial Multipath Fading Processes
We analyse the spatial diversity of a multipath fading process for a finite
region or curve in the plane. By means of the Karhunen-Lo\`eve (KL) expansion,
this diversity can be characterised by the eigenvalue spectrum of the spatial
autocorrelation kernel. This justifies to use the term diversity spectrum for
it. We show how the diversity spectrum can be calculated for any such
geometrical object and any fading statistics represented by the power azimuth
spectrum (PAS). We give rigorous estimates for the accuracy of the numerically
calculated eigenvalues. The numerically calculated diversity spectra provide
useful hints for the optimisation of the geometry of an antenna array.
Furthermore, for a channel coded system, they allow to evaluate the time
interleaving depth that is necessary to exploit the diversity gain of the code.Comment: 32 pages, 10 figure
Periodic orbits for space-based reflectors in the circular restricted three-body problem
The use of space-based orbital reflectors to increase the total insolation of the Earth has been considered with potential applications in night-side illumination, electric power generation and climate engineering. Previous studies have demonstrated that families of displaced Earth-centered and artificial halo orbits may be generated using continuous propulsion, e.g. solar sails. In this work, a three-body analysis is performed by using the circular restricted three body problem, such that, the space mirror attitude reflects sunlight in the direction of Earth’s center, increasing the total insolation. Using the Lindstedt–Poincaré and differential corrector methods, a family of halo orbits at artificial Sun–Earth L2 points are found. It is shown that the third order approximation does not yield real solutions after the reflector acceleration exceeds 0.245 mm s−2, i.e. the analytical expressions for the in- and out-of-plane amplitudes yield imaginary values. Thus, a larger solar reflector acceleration is required to obtain periodic orbits closer to the Earth. Derived using a two-body approach and applying the differential corrector method, a family of displaced periodic orbits close to the Earth are therefore found, with a solar reflector acceleration of 2.686 mm s−2
Central limit theorems for the radial spanning tree
Consider a homogeneous Poisson point process in a compact convex set in
-dimensional Euclidean space which has interior points and contains the
origin. The radial spanning tree is constructed by connecting each point of the
Poisson point process with its nearest neighbour that is closer to the origin.
For increasing intensity of the underlying Poisson point process the paper
provides expectation and variance asymptotics as well as central limit theorems
with rates of convergence for a class of edge functionals including the total
edge length
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