Description Usage Arguments Details Value References Examples
The 1D algorithm to estimate the envelope subspace based on the line search algorithm for optimization on manifold. The line search 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. |
... |
Additional user-defined arguments for the line search algorithm:
The default values are: |
The objective function F(w) and its gradient G(w) in line search algorithm are:
F(w)=\log|w^T M_k w|+\log|w^T(M_k+U_k)^{-1}w|
G(w) = dF/dw = 2 (w^T M_k w)^{-1} M_k w + 2 (w^T (M_k + U_k)^{-1} w)^{-1}(M_k + U_k)^{-1} w
See Cook, R. D., & Zhang, X. (2016) for more details of the 1D algorithm.
Return the estimated orthogonal basis of the envelope subspace.
Cook, R.D. and Zhang, X., 2016. Algorithms for envelope estimation. Journal of Computational and Graphical Statistics, 25(1), pp.284-300.
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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