Exponential map of \(Gr(n, k)\) at base point \(U_0\), the inverse
of log_map_grassmann on the cut domain. Each row of
V holds tangent coordinates as.vector(B), with B of
size (n - k) x k; the horizontal lift is
\(\Delta = U_\perp B\) and, writing its compact singular value
decomposition \(\Delta = Q \Sigma R^T\), the point is
\(U_0 R \cos(\Sigma) R^T + Q \sin(\Sigma) R^T\). Useful to simulate
data on \(Gr(n, k)\).
Arguments
- V
Numeric matrix with
(n - k) * kcolumns: tangent-space coordinates atp, one row per point.- p
The base point: an
nxkmatrix with orthonormal columns, or a vector of lengthn * k(thennandkmust be given).- n, k
Integers: dimensions of \(Gr(n, k)\). Default to the dimensions of
pwhen it is a matrix.
Value
A numeric matrix with nrow(V) rows and n * k
columns: Stiefel representatives as.vector(U) of the points.
Examples
X <- exp_map_grassmann(matrix(rnorm(40, sd = 0.3), nrow = 10), diag(4)[, 1:2])
is_in_grassmann(X, n = 4, k = 2)
#> [1] TRUE