rnormtangents {GeodRegr}

Random generation of tangent vectors from the Riemannian normal distribution

Description

Random generation of tangent vectors from the Riemannian normal distribution on the n-dimensional sphere or hyperbolic space at mean (1, 0, ..., 0), a vector of length n+1.

Usage

rnormtangents(manifold, N, n, sigma_sq)

Arguments

manifold

Type of manifold ('sphere' or 'hyperbolic').

N

Number of points to generate.

n

Dimension of the manifold.

sigma_sq

A scale parameter.

Details

Tangent vectors are of the form \mathrm{Log}(\mu, y) in the tangent space at the Fr\'echet mean \mu = (1, 0, ..., 0), which is isomorphic to n-dimensional Euclidean space, where y has a Riemannian normal distribution. The first element of these vectors will always be 0 at this \mu. These vectors can be transported to any other \mu on the manifold.

Value

An (n+1)-by-N matrix where each column represents a random tangent vector at (1, 0, ..., 0).

Author(s)

Ha-Young Shin

References

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

Fletcher, T. (2020). Statistics on manifolds. In Riemannian Geometric Statistics in Medical Image Analysis. 39–74. Academic Press.

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

Examples

sims <- rnormtangents('hyperbolic', N = 4, n = 2, sigma_sq = 1)