Description Usage Arguments Details Value References See Also Examples
The FG algorithm to estimate the envelope subspace based on the curvilinear search algorithm for optimization on Stiefel manifold. The curvilinear algorithm is developed by Wen and Yin (2013) and the Matlab version is implemented in the Matlab package OptM.
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M |
The p-by-p positive definite matrix M in the envelope objective function. |
U |
The p-by-p positive semi-definite matrix U in the envelope objective function. |
u |
An integer between 0 and n representing the envelope dimension. Ignored if |
Gamma_init |
Initial envelope subspace basis. The default value is the estimator from |
... |
Additional user-defined arguments for the curvilinear search algorithm:
The default values are: |
If Gamma_init
is provided, then the envelope dimension u = ncol(Gamma_init)
.
The function OptMFG
calls the function OptStiefelGBB
internally which implements the curvilinear search algorithm.
The objective function F(Γ) and its gradient G(Γ) in curvilinear search algorithm are:
F(Γ)=\log|Γ^T M Γ|+\log| Γ^T(M+U)^{-1}Γ|
G(Γ) = dF/d Γ = 2 M Γ (Γ^T M Γ)^{-1} + 2 (M + U)^{-1} Γ (Γ^T (M + U)^{-1} Γ)^{-1}
Return the estimated orthogonal basis of the envelope subspace.
Wen, Z. and Yin, W., 2013. A feasible method for optimization with orthogonality constraints. Mathematical Programming, 142(1-2), pp.397-434.
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