
Riemannian Geometry of Symmetric Positive Definite Matrices via Cholesky Decomposition
We present a new Riemannian metric, termed LogCholesky metric, on the m...
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Parallel Transport with Pole Ladder: a Third Order Scheme in Affine Connection Spaces which is Exact in Affine Symmetric Spaces
Parallel transport is an important step in many discrete algorithms for ...
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Additive Models for Symmetric PositiveDefinite Matrices, Riemannian Manifolds and Lie groups
In this paper an additive regression model for a symmetric positivedefi...
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Numerical approximation of boundary value problems for curvature flow and elastic flow in Riemannian manifolds
We present variational approximations of boundary value problems for cur...
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Symmetric Algorithmic Components for Shape Analysis with Diffeomorphisms
In computational anatomy, the statistical analysis of temporal deformati...
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Principal Autoparallel Analysis: Data Analysis in Weitzenböck Space
The statistical analysis of data lying on a differentiable, locally Eucl...
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Convergence analysis of subdivision processes on the sphere
This paper provides a strategy to analyse the convergence of nonlinear a...
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Numerical Accuracy of Ladder Schemes for Parallel Transport on Manifolds
Parallel transport is a fundamental tool to perform statistics on Riemannian manifolds. Since closed formulae don't exist in general, practitioners often have to resort to numerical schemes. Ladder methods are a popular class of algorithms that rely on iterative constructions of geodesic parallelograms. And yet, the literature lacks a clear analysis of their convergence performance. In this work, we give Taylor approximations of the elementary constructions of Schild's ladder and the pole ladder with respect to the Riemann curvature of the underlying space. We then prove that these methods can be iterated to converge with quadratic speed, even when geodesics are approximated by numerical schemes. We also contribute a new link between Schild's ladder and the Fanning Scheme which explains why the latter naturally converges only linearly. The extra computational cost of ladder methods is thus easily compensated by a drastic reduction of the number of steps needed to achieve the requested accuracy. Illustrations on the 2sphere, the space of symmetric positive definite matrices and the special Euclidean group show that the theoretical errors we have established are measured with a high accuracy in practice. The special Euclidean group with an anisotropic leftinvariant metric is of particular interest as it is a tractable example of a nonsymmetric space in general , which reduces to a Riemannian symmetric space in a particular case. As a secondary contribution, we compute the covariant derivative of the curvature in this space.
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