92 research outputs found
Shape analysis on homogeneous spaces: a generalised SRVT framework
Shape analysis is ubiquitous in problems of pattern and object recognition
and has developed considerably in the last decade. The use of shapes is natural
in applications where one wants to compare curves independently of their
parametrisation. One computationally efficient approach to shape analysis is
based on the Square Root Velocity Transform (SRVT). In this paper we propose a
generalised SRVT framework for shapes on homogeneous manifolds. The method
opens up for a variety of possibilities based on different choices of Lie group
action and giving rise to different Riemannian metrics.Comment: 28 pages; 4 figures, 30 subfigures; notes for proceedings of the Abel
Symposium 2016: "Computation and Combinatorics in Dynamics, Stochastics and
Control". v3: amended the text to improve readability and clarify some
points; updated and added some references; added pseudocode for the dynamic
programming algorithm used. The main results remain unchange
Deep learning as optimal control problems: Models and numerical methods
We consider recent work of Haber and Ruthotto 2017 and Chang et al. 2018,
where deep learning neural networks have been interpreted as discretisations of
an optimal control problem subject to an ordinary differential equation
constraint. We review the first order conditions for optimality, and the
conditions ensuring optimality after discretisation. This leads to a class of
algorithms for solving the discrete optimal control problem which guarantee
that the corresponding discrete necessary conditions for optimality are
fulfilled. The differential equation setting lends itself to learning
additional parameters such as the time discretisation. We explore this
extension alongside natural constraints (e.g. time steps lie in a simplex). We
compare these deep learning algorithms numerically in terms of induced flow and
generalisation ability
Deep learning as optimal control problems
We briefly review recent work where deep learning neural networks have been interpreted as discretisations of an optimal control problem subject to an ordinary differential equation constraint. We report here new preliminary experiments with implicit symplectic Runge-Kutta methods. In this paper, we discuss ongoing and future research in this area
Using discrete Darboux polynomials to detect and determine preserved measures and integrals of rational maps
In this Letter we propose a systematic approach for detecting and calculating
preserved measures and integrals of a rational map. The approach is based on
the use of cofactors and Discrete Darboux Polynomials and relies on the use of
symbolic algebra tools. Given sufficient computing power, all rational
preserved integrals can be found.
We show, in two examples, how to use this method to detect and determine
preserved measures and integrals of the considered rational maps.Comment: 8 pages, 1 Figur
Structure-preserving deep learning
Over the past few years, deep learning has risen to the foreground as a topic
of massive interest, mainly as a result of successes obtained in solving
large-scale image processing tasks. There are multiple challenging mathematical
problems involved in applying deep learning: most deep learning methods require
the solution of hard optimisation problems, and a good understanding of the
tradeoff between computational effort, amount of data and model complexity is
required to successfully design a deep learning approach for a given problem. A
large amount of progress made in deep learning has been based on heuristic
explorations, but there is a growing effort to mathematically understand the
structure in existing deep learning methods and to systematically design new
deep learning methods to preserve certain types of structure in deep learning.
In this article, we review a number of these directions: some deep neural
networks can be understood as discretisations of dynamical systems, neural
networks can be designed to have desirable properties such as invertibility or
group equivariance, and new algorithmic frameworks based on conformal
Hamiltonian systems and Riemannian manifolds to solve the optimisation problems
have been proposed. We conclude our review of each of these topics by
discussing some open problems that we consider to be interesting directions for
future research
Probing magnetic fields with multi-frequency polarized synchrotron emission
We investigate the problem of probing the local spatial structure of the
magnetic field of the interstellar medium using multi-frequency polarized maps
of the synchrotron emission at radio wavelengths. We focus in this paper on the
three-dimensional reconstruction of the largest scales of the magnetic field,
relying on the internal depolarization (due to differential Faraday rotation)
of the emitting medium as a function of electromagnetic frequency. We argue
that multi-band spectroscopy in the radio wavelengths, developed in the context
of high-redshift extragalactic HI lines, can be a very useful probe of the 3D
magnetic field structure of our Galaxy when combined with a Maximum A
Posteriori reconstruction technique. When starting from a fair approximation of
the magnetic field, we are able to recover the true one by using a linearized
version of the corresponding inverse problem. The spectral analysis of this
problem allows us to specify the best sampling strategy in electromagnetic
frequency and predicts a spatially anisotropic distribution of posterior
errors. The reconstruction method is illustrated for reference fields extracted
from realistic magneto-hydrodynamical simulations
Exact Analytic Solution for the Rotation of a Rigid Body having Spherical Ellipsoid of Inertia and Subjected to a Constant Torque
The exact analytic solution is introduced for the rotational motion of a
rigid body having three equal principal moments of inertia and subjected to an
external torque vector which is constant for an observer fixed with the body,
and to arbitrary initial angular velocity. In the paper a parametrization of
the rotation by three complex numbers is used. In particular, the rows of the
rotation matrix are seen as elements of the unit sphere and projected, by
stereographic projection, onto points on the complex plane. In this
representation, the kinematic differential equation reduces to an equation of
Riccati type, which is solved through appropriate choices of substitutions,
thereby yielding an analytic solution in terms of confluent hypergeometric
functions. The rotation matrix is recovered from the three complex rotation
variables by inverse stereographic map. The results of a numerical experiment
confirming the exactness of the analytic solution are reported. The newly found
analytic solution is valid for any motion time length and rotation amplitude.
The present paper adds a further element to the small set of special cases for
which an exact solution of the rotational motion of a rigid body exists.Comment: "Errata Corridge Postprint" In particular: typos present in Eq. 28 of
the Journal version are HERE correcte
Exact Analytic Solutions for the Rotation of an Axially Symmetric Rigid Body Subjected to a Constant Torque
New exact analytic solutions are introduced for the rotational motion of a
rigid body having two equal principal moments of inertia and subjected to an
external torque which is constant in magnitude. In particular, the solutions
are obtained for the following cases: (1) Torque parallel to the symmetry axis
and arbitrary initial angular velocity; (2) Torque perpendicular to the
symmetry axis and such that the torque is rotating at a constant rate about the
symmetry axis, and arbitrary initial angular velocity; (3) Torque and initial
angular velocity perpendicular to the symmetry axis, with the torque being
fixed with the body. In addition to the solutions for these three forced cases,
an original solution is introduced for the case of torque-free motion, which is
simpler than the classical solution as regards its derivation and uses the
rotation matrix in order to describe the body orientation. This paper builds
upon the recently discovered exact solution for the motion of a rigid body with
a spherical ellipsoid of inertia. In particular, by following Hestenes' theory,
the rotational motion of an axially symmetric rigid body is seen at any instant
in time as the combination of the motion of a "virtual" spherical body with
respect to the inertial frame and the motion of the axially symmetric body with
respect to this "virtual" body. The kinematic solutions are presented in terms
of the rotation matrix. The newly found exact analytic solutions are valid for
any motion time length and rotation amplitude. The present paper adds further
elements to the small set of special cases for which an exact solution of the
rotational motion of a rigid body exists.Comment: "Errata Corridge Postprint" version of the journal paper. The
following typos present in the Journal version are HERE corrected: 1)
Definition of \beta, before Eq. 18; 2) sign in the statement of Theorem 3; 3)
Sign in Eq. 53; 4)Item r_0 in Eq. 58; 5) Item R_{SN}(0) in Eq. 6
Differential Geometry applied to Acoustics : Non Linear Propagation in Reissner Beams
Although acoustics is one of the disciplines of mechanics, its
"geometrization" is still limited to a few areas. As shown in the work on
nonlinear propagation in Reissner beams, it seems that an interpretation of the
theories of acoustics through the concepts of differential geometry can help to
address the non-linear phenomena in their intrinsic qualities. This results in
a field of research aimed at establishing and solving dynamic models purged of
any artificial nonlinearity by taking advantage of symmetry properties
underlying the use of Lie groups. The geometric constructions needed for
reduction are presented in the context of the "covariant" approach.Comment: Submitted to GSI2013 - Geometric Science of Informatio
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