Tangent-Space Distance Matrix

Compute a distance matrix after mapping Lorentz hyperboloid points to T_mu H^2 with the logarithmic map. Distances are computed using the Lorentz inner product restricted to the tangent space, which is positive definite. Equivalently, this is Euclidean distance in a Lorentz-orthonormal tangent basis.

Usage

tangent_distance_matrix(Z, mu = NULL, tol = 1e-10)

Arguments

Z

n x 3 Lorentz hyperboloid coordinates.

mu

Reference point. If NULL, the normalized extrinsic mean is used.

tol

Intrinsic-distance threshold used by lorentz_log_map(); points within this distance of mu are mapped to zero.

Value

A symmetric n x n tangent-space distance matrix.

Details

With \(v_i=\log_\mu(z_i)\) and \(v_j=\log_\mu(z_j)\), the distance is $$d_{T,\mu}(z_i,z_j)=\sqrt{\langle v_i-v_j,v_i-v_j\rangle_L}.$$ Here \(i,j\) index input rows and lorentz_inner() defines the inner product. Its restriction to the common tangent plane is positive definite. These distances define the Tangent-E weighting approximation: radial distances from mu are preserved in exact arithmetic, but general pairwise hyperbolic distances are not. The implementation uses coordinate differences and gives identical mapped points exactly zero distance. The tol argument applies to each point's distance from mu in the log map, not to arbitrary pairwise distances between distinct tangent coordinates.

Examples

H <- simulate_H2(5, r_max = 2)
tangent_distance_matrix(H$Z)