geo_reg {GeodRegr}

Gradient descent for (robust) geodesic regression

Description

Finds \mathrm{argmin}_{(p,V)\in M\times (T_pM) ^ n}\sum_{i=1} ^ {N} \rho(d(\mathrm{Exp}(p,Vx_i),y_i)) through a gradient descent algorithm.

Usage

geo_reg(
  manifold,
  x,
  y,
  estimator,
  c = NULL,
  p_tol = 1e-05,
  V_tol = 1e-05,
  max_iter = 1e+05
)

Arguments

manifold

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

x

A vector, matrix, or data frame of independent variables; for matrices and data frames, the rows and columns represent the subjects and independent variables, respectively.

y

A matrix or data frame whose columns represent points on the manifold.

estimator

M-type estimator ('l2', 'l1', 'huber', or 'tukey').

c

Multiplier of \sigma, the square root of the variance, used in the cutoff parameter for the 'huber' and 'tukey' estimators; should be NULL for the 'l2' or 'l1' estimators.

p_tol

Termination condition for the distance between consecutive updates of p.

V_tol

Termination condition for the distance between columns of consecutive updates of V, parallel transported to be in the same tangent space. Can be a vector of positive real numbers for each independent variable or a single positive number.

max_iter

Maximum number of gradient descent steps before ending the algorithm.

Details

Each column of x should be centered to have an average of 0 for the quickest and most accurate results. If all of the elements of a column of x are equal, the resulting vector will consist of NAs. In the case of the 'sphere', an error will be raised if all points are on a pair of antipodes.

Value

A named list containing

p

a vector representing the estimate of the initial point on the manifold

V

a matrix representing the estimate of the initial velocities for each independent variable; the columns represent the independent variables.

iteration

number of gradient descent steps taken.

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.

Kim, H. J., Adluru, N., Collins, M. D., Chung, M. K., Bendin, B. B., Johnson, S. C., Davidson, R. J. and Singh, V. (2014). Multivariate general linear models (MGLM) on Riemannian manifolds with applications to statistical analysis of diffusion weighted images. 2014 IEEE Conference on Computer Vision and Pattern Recognition, 2705-2712.

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

See Also

intrinsic_location.

Examples

# an example of multiple regression with two independent variables, with 64
# data points

x <- matrix(runif(2 * 64), ncol = 2)
x <- t(t(x) - colMeans(x))
y <- matrix(0L, nrow = 4, ncol = 64)
for (i in 1:64) {
  y[, i] <- exp_map('sphere', c(1, 0, 0, 0), c(0, runif(1), runif(1),
      runif(1)))
}
geo_reg('sphere', x, y, 'tukey', c = are_nr('tukey', 2, 6))