Find a Grassmann base point via multi-start optimization.

Grassmann case of optimize_base_point: numerically minimizes the total Grassmann distance (dist_grassmann) from a candidate point to every row of X. The search is unconstrained, over the tangent space at the projection mean (auto_base_point_grassmann), with restarts drawn from a Gaussian scaled to the spread of the data. See choose_base_point.

Usage

optimize_base_point_grassmann(X, n, k, n_start = 10, maxit = 200, seed = 42)

Arguments

X

A matrix or data frame of Grassmann data; rows are samples, with n * k columns (see manifold_grassmann).

n

Integer: dimension of the ambient space.

k

Integer: dimension of the subspaces, 1 <= k <= n - 1.

n_start

Number of random restarts used to avoid local minima of the (non-convex) objective. Defaults to 10.

maxit

Maximum number of optim iterations per restart. Defaults to 200.

seed

Integer random seed used to generate the restart points, for reproducibility. Defaults to 42.

Value

A list with three elements: p (the best point found across all restarts, as a vector of length n * k), value (its total Grassmann distance to the rows of X), and convergence (the optim convergence code of the best restart; 0 indicates successful convergence).

Examples

X <- exp_map_grassmann(matrix(rnorm(40, sd = 0.3), nrow = 10), diag(4)[, 1:2])
optimize_base_point_grassmann(X, n = 4, k = 2, n_start = 3)
#> $p
#> [1]  0.969052410 -0.209518770 -0.041681316  0.123701173 -0.223729175
#> [6] -0.967802495  0.005286253  0.115219972
#> 
#> $value
#> [1] 4.635188
#> 
#> $convergence
#> [1] 0
#>