dot-rotate_simplimax_oblq: Oblique simplimax factor rotation

In EFAtools: Fast and Flexible Implementations of Exploratory Factor Analysis Tools

Oblique simplimax factor rotation

Description

Rotate a loading matrix obliquely under the simplimax criterion using a gradient-projection optimizer along the oblique (column-normalized) manifold.

Usage

.rotate_simplimax_oblq(
  L,
  k,
  eps = 1e-05,
  normalize = TRUE,
  random_starts = 0L,
  maxit = 1000L,
  max_line_search = 10L,
  step0 = 1
)

Arguments

L	Numeric matrix. The unrotated loading matrix (variables by factors).
k	Integer scalar. The number of "close-to-zero" loadings the criterion targets; must be in ⁠[1, nrow(L) * ncol(L)]⁠. k = nrow(L) is the usual default.
eps	Numeric scalar. Convergence tolerance for the projected-gradient norm. Because the simplimax criterion is only piecewise smooth, the projected gradient need not reach this tolerance at the optimum; convergence is then reported when the criterion value stalls (the non-monotone search described above), so eps mainly governs the smooth phases of the search.
normalize	Logical scalar. If TRUE, apply Kaiser normalization before rotation and reverse it afterwards.
random_starts	Integer scalar. Number of random orthogonal starts fully optimized in addition to the identity start.
maxit	Integer scalar. Maximum number of projected-gradient updates per start.
max_line_search	Integer scalar. Maximum number of step-halving attempts after the initial trial step in each line-search phase.
step0	Numeric scalar. Initial step size used in the projected-gradient update.

Details

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 simplimax criterion sums the k smallest squared loadings, so it is minimized when the k "close-to-zero" loadings are driven toward zero; the count k is a tuning parameter. Because the set of k smallest loadings is reselected at every evaluation, the criterion is only piecewise smooth: its gradient jumps as loadings cross the kth-smallest threshold, so the line search accepts a step whenever it decreases the largest objective over a short window of recent iterations (a non-monotone test; Grippo, Lampariello, & Lucidi, 1986), letting the optimizer step across the kinks where a strictly monotone descent would stall.

The criterion is strongly prone to local minima, so the solver fully optimizes the identity start together with random_starts random orthogonal starts and keeps the solution with the lowest criterion value. Fully optimizing every start – rather than the screen-and-triage strategy used for the smooth criteria, which assumes the rational start lies in the global basin – is the standard remedy for the local minima of complexity-based rotation criteria (Kiers, 1994; Browne, 2001).

Value

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.

References

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.

Grippo, L., Lampariello, F., & Lucidi, S. (1986). A nonmonotone line search technique for Newton's method. SIAM Journal on Numerical Analysis, 23, 707-716.

Kiers, H. A. L. (1994). Simplimax: Oblique rotation to an optimal target with simple structure. Psychometrika, 59, 567-579.

EFAtools documentation built on Aug. 21, 2026, 5:16 p.m.