Find a hyperbolic base point via the Lorentzian centroid.

Closed-form, data-centered base point for \(H^d\): the Euclidean mean \(m\) of the rows of X, rescaled back onto the hyperboloid, \(p = m / \sqrt{-\langle m, m \rangle_L}\). The mean of points on the upper sheet is always future timelike, so p is well defined. It is the minimizer of the sum of squared Lorentzian distances to the data, and is close to the Fréchet mean when the data are concentrated.

Usage

auto_base_point_hyperbolic(X)

Arguments

X

A matrix or data frame of points on the hyperboloid; rows are samples, with d + 1 columns.

Value

Numeric vector of length ncol(X): the chosen base point.

Examples

X <- exp_map_hyperbolic(matrix(rnorm(20), nrow = 10, ncol = 2))
auto_base_point_hyperbolic(X)
#> [1]  1.0624557  0.1059847 -0.3428986