Center-outward transform.

Maps d-dimensional Euclidean data to the open unit ball \(S^d\), following Hallin et al. (2021), by returning the empirical center-outward distribution function \(F^{(n)}: \mathbb{R}^d \to S^d\): an n x d matrix whose rows are the grid values (build_ball_grid) assigned to each observation by the optimal \(L^2\) coupling (optimal_assignment). \(F\) is bounded in the unit ball, so any test statistic computed on \(F\) (e.g. a distance covariance test) requires no moment condition on the original data.

Usage

center_outward_transform(Z, seed = 42)

Arguments

Z

Numeric matrix (n x d): Euclidean data, typically the output of a manifold log map (e.g. log_map_torus).

seed

Integer random seed passed to build_ball_grid for the grid's quasi-uniform directions (only used when d > 2). Defaults to 42.

Value

Numeric matrix F (n x d): rows inside the unit ball \(S^d\), the multidimensional rank of each row of Z.

Details

This is the transform behind the RankCov test of Liu et al. (2026), run by manifold.dcov.test with center_outward = TRUE: the distance covariance test is applied to the center-outward ranks of the tangent-space coordinates.

References

Liu, H., Li, M., González-Delgado, J. and Hallin, M. (2026). Independence test on Riemannian manifolds. Manuscript.

Hallin, M., del Barrio, E., Cuesta-Albertos, J. and Matrán, C. (2021). Distribution and quantile functions, ranks and signs in dimension d: a measure transportation approach. The Annals of Statistics, 49(2), 1139–1165.

Examples

Z <- matrix(rnorm(20), nrow = 10, ncol = 2)
F <- center_outward_transform(Z)
dim(F)
#> [1] 10  2
max(sqrt(rowSums(F^2))) <= 1
#> [1] TRUE