Rotate a loading matrix obliquely under the geomin criterion using a gradient-projection optimizer along the oblique (column-normalized) manifold.
.rotate_geomin_oblq(
L,
delta = 0.01,
eps = 1e-05,
normalize = TRUE,
random_starts = 0L,
maxit = 1000L,
max_line_search = 10L,
step0 = 1,
screen_keep = 5L,
triage_maxit = 25L,
triage_improve_tol = 0
)A named list with the rotated loadings, the transformation matrix Th
(with L %*% t(solve(Th)) reproducing the rotated loadings), the factor correlation
matrix Phi (t(Th) %*% Th), the attained criterion value, and the convergence and
validity flags. The list additionally reports the criterion value reached
at each optimized start in all_values, with a per-start convergence flag in all_converged.
Numeric matrix. The unrotated loading matrix (variables by factors).
Numeric scalar. The geomin offset added to the squared loadings; must be a
positive finite scalar. delta = 0.01 is the usual default.
Numeric scalar. Convergence tolerance for the projected-gradient norm.
Logical scalar. If TRUE, apply Kaiser normalization before
rotation and reverse it afterwards.
Integer scalar. Number of additional random orthogonal starts.
Integer scalar. Maximum number of projected-gradient updates.
Integer scalar. Maximum number of step-halving attempts after the initial trial step in each line-search phase.
Numeric scalar. Initial step size used in the projected-gradient update.
Integer scalar. Number of screened random starts retained for triage optimization.
Integer scalar. Number of short optimization iterations used in the triage stage.
Numeric scalar. Relative improvement required for a triaged start to be promoted to full optimization.
The criterion value f and its gradient dQ/dL at the rotated loadings
L = A %*% solve(t(T)) define the search; the engine maps the gradient to the
transformation T on the manifold diag(t(T) %*% T) = 1, projects it onto the tangent
space, performs a non-monotone line search, and retracts back onto the manifold by
column normalization. The geomin criterion sums the per-variable geometric mean of the
squared loadings offset by delta; it is prone to local minima, so additional random
starts are recommended.
Additional random starts may be requested. To bound runtime the solver screens each random
start by its objective, runs a short triage optimization on the best-screened starts, and
fully optimizes only those that improve on the current incumbent by at least
triage_improve_tol.
Bernaards, C. A., & Jennrich, R. I. (2005). Gradient projection algorithms and software for arbitrary rotation criteria in factor analysis. Educational and Psychological Measurement, 65, 676-696.
Browne, M. W. (2001). An overview of analytic rotation in exploratory factor analysis. Multivariate Behavioral Research, 36, 111-150.