GeodRegr (version 0.2.0)

log_map: Logarithm map

Description

Performs the logarithm map \(\textrm{Log}_{p_1}(p_2)\) on the given manifold, provided \(p_2\) is in the domain of \(\textrm{Log}_{p_1}\).

Usage

log_map(manifold, p1, p2)

Arguments

manifold

Type of manifold ('euclidean', 'sphere', 'hyperbolic', or 'kendall').

p1

A vector (or column matrix) representing a point on the manifold.

p2

A vector (or column matrix) representing a point on the manifold.

Value

A vector tangent to p1.

Details

On the sphere, \(-p_1\) is not in the domain of \(\textrm{Log}_{p_1}\).

References

Fletcher, P. T. (2013). Geodesic regression and the theory of least squares on Riemannian manifolds. International Journal of Computer Vision, 105, 171-185.

Cornea, E., Zhu, H., Kim, P. and Ibrahim, J. G. (2017). Regression models on Riemannian symmetric spaces. Journal of the Royal Statistical Society: Series B, 79, 463-482.

Calinon, S. (2020). Gaussians on Riemannian manifolds: Applications for robot learning and adaptive control. IEEE Robotics & Automation Magazine, 27, 33-45.

Shin, H.-Y. and Oh H.-S. (2020). Robust Geodesic Regression. <arXiv:2007.04518>

See Also

exp_map, geo_dist.

Examples

# NOT RUN {
log_map('sphere', c(0, 1, 0, 0), c(0, 0, 1, 0))

# }